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\put(-40,200){\sx{2}{\begin{parbox}{250pt}{\flushright 
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Non-integer iterates of holomorphic functions. \\  Tetration and other superfunctions. Formulas,\\  algorithms, tables, graphics and complex maps.}\end{parbox}}   }
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%\url{https://mizugadro.mydns.jp/t/index.php/File:Tetma.jpg}  cover map\\
%\url{https://mizugadro.mydns.jp/BOOK/444.pdf}\\
%\url{https://www.ils.uec.ac.jp/~dima/BOOK/443.pdf} (previous)
%\url{https://mizugadro.mydns.jp/t/index.php/Book}

\vskip 3pt
\sx{.98}{Dmitrii Kouznetsov. Superfunctions. Lambert Academoc Publishing, 2020}

\url{https://mizugadro.mydns.jp/BOOK/444.pdf}
\vskip 2pt

This version %is under construction, 
generated from Latex source \today

\vskip 14pt

The Russian version of year 2014 (Fig.\ref{LepaDim}), Isbn: 978-3-659-56202-0
\vskip -30pt
~
{\normalsize
\begin{verbatim}
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
\end{verbatim}
}\vskip -12pt
%{\small(The Russian title is "Суперфункции", but the Russian characters appear as abrakadabra at copypast)}\\
List of misprints in the First edition of the Russian version is at \\
\url{https://mizugadro.mydns.jp/t/index.php/Superfunctions_in_Russian}\\
I load the corrected Russian version as \\
\url{https://mizugadro.mydns.jp/BOOK/202.pdf}\\
\url{https://www.ils.uec.ac.jp/~dima/BOOK/202.pdf}
\vskip 18pt

List of figures (for both English and versions):\\
\url{https://mizugadro.mydns.jp/t/index.php/Category:Book}

\begin{figure}[h]
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\begin{center}
\sx{.55}{\ing{LepaDim}}
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\caption{Distribution of the Russian version of this Book, 2014 \label{LepaDim}}
\vskip -18pt
\end{figure}

\baselineskip 17.7pt
\tableofcontents 
\vskip 12pt

{\bf \sx{1.06}{References} \rule{316pt}{0pt} \pageref{references}} 

\pagestyle{plain}
\large
\chapter{~ Prologue}
\baselineskip 17pt

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\caption{Cloud in Trousers$\!\!$\label{oblako}}
\vskip -8pt
\end{wrapfigure}

I type this Prologue, respecting my teachers of English, and also those of the Russian Literature.
They think,
that every scribbling, % (even if it is not in the classical English), 
since W.Shakespeare,
should begin with some Prologue\footnote{ W. Shakespear. Romeo and Juliet. Prologue. 932/Sh12, Kenkyusba English Classics, 1926.},
and should finish with some Epilogue. In addition, the Soviet teachers had promoted the sovietism\footnote{\url{https://mizugadro.mydns.jp/t/index.php/Sovetism}}
that even a cloud should wear some 
trousers\footnote{\url{https://www.unlikelystories.org/old/archives/cloudintrousers.html} A Cloud in Trousers by Vladimir Mayakovsky, translated from the Russian by Andrey Kneller. Prologue. Cited by the state for 2014.09.10.}, as it is shown in Fig.\ref{oblako}.
\vskip 3pt
This prologue %should be considered
appears as a kind of trousers, a wrap, to cover the inside; where, my results are presented in a 
naked and provocative form.
\vskip 6pt

This Book is about holomorphic (analytic) solutions $F$ of equation 
\be
F(z\!+\!1)=T(F(z))
\iL{prologeq}
\ee
where $T$ is some given holomorphic (analytic) function. Also, this Book is about applications of the solution $F$, its inverse function $G\!=\!F^{-1}$ and interesting properties of 
function $T^n(z)\!=\!F(n\!+\! G(z))$.
The given function $T$ is called ``transfer function''; the solution $F$ is its ``superfunction''.
The inverse function $G$ is called the ``Abelfunction''.
\vskip 6pt

The Prologue describes not functions $F$ and $G$ above, but this Book: what is it, why is it, how to use it, what is in it, and what is not. 
I follow the classical example\footnote{ 
François Rabelais. %, translated by Thomas Urquhart and Peter Antony Motteux. 
Gargantua \cite{Rabelais} : %/The Author's Prologue to the First Book.\ref{Rabelais}:
.. Most noble and illustrious drinkers, and you thrice precious pockified blades (for to you, and none else, do I dedicate my writings), %Alcibiades, in that dialogue of Plato's, which is entitled The Banquet, whilst he was setting forth the praises of his schoolmaster Socrates (without all question the prince of philosophers), amongst other discourses to that purpose, said that he resembled the Silenes.
}
and explain, why this Book is so important.
\vskip 4pt

Readers, who are interested namely in solutions of the equation above and their physical meaning, may scroll from here directly to Introduction,
Chapter \ref{intro} at page \pageref{intro}.

\section{For whom is this Prologue}
\baselineskip 17.6pt

I type this Prologue not for the Readers. I type this Prologue for those, who will not read this Book:
\vskip 1mm

{\bf For Editors}, managers, sellers, who deal with thousands books,
and who have need to understand, within a minute, why namely this Book should
be placed at the top in the list of recommended literature.
\vskip 1mm

{\bf For the experienced critics}, who read only two or three pages, in order to write the review:{\em 
What oddity is this:
Superfunctions? What sort of Superfunctions have we here? And thrust into the world by a laserist! 
As if we are not satisfied with superconductivity, supersymmetry, superfluidity, supermen and supermarkets!
As though threes enough had not been cut for paper, and not enough files have been loaded into internet! As though folks enough of all classes had not tired their fingers with keyboards! The whim must take a laserist to follow their example! Really there is such a lot of paper nowadays that it takes time to think what to wrap in it!}
\footnote{%\url{https://xixvek.wordpress.com/2013/03/27/translation-comparison-evenings-at-a-farmhouse-near-dikanka/}
N.Gogol. Evenings at a Farmhouse near Dikanka. MARCH 27, 2013. \cite{xutore} :~
“What oddity is this: Evenings on a Farm near Dikanka?
What sort of Evenings have we here? And thrust into the world by a beekeeper!
God protect us! As though geese enough had not been plucked for pens and rags turned into paper!
As though folks enough of all classes had not covered their fingers with inkstains!
The whim must take a beekeeper to follow their example! 
Really there is such a lot of paper nowadays that it takes time to think what to wrap in it.”}
\vskip 1mm

{\bf For librarians}, who need to find for this Book a suitable place, in order to make it visible at the background of tons of books about superconductivity, supermen, supermarkets and superinflation.
\vskip 1mm

{\bf For colleagues}, 
who may wonder, why the laserist, instead of to promote the optical ceramics (for which the big Grants are assigned)
deals with superfunctions (which are yet very far from to get the financial support), and behaves as a  simpleton, who does not know, 
{\em How to Win Friends and Influence People}.
\vskip 1mm

{\bf For relatives} and friends, who teach me to live and to promote my results. They seem to know very well, how to 
{\em influence people}. This book could be a kind of medicine against this kind of influenza.
However, more detailed analysis of relations between friends is published separately
\cite{klavanur}.
\vskip 2mm

For the categories of non-readers mentioned above, I should specify the genre and topic of this book. 
The next two sections are dedicated to this. 
With such a specification, one may form his/her/its own opinion about the book, without to read it.

\section{Russian version and copyleft}
\baselineskip 18pt

\begin{wrapfigure}{r}{200px}
\sx{.33}{\begin{picture}(400,120)
\put(40,-30){\ing{NakedConductor}}
\end{picture}}
\vskip -2pt
\caption{Problem of translation \label{conductor}}
\vskip -8pt
\end{wrapfigure}

This book had been planned as the English version of the Russian book ``Суперфункции''
\footnote{\url{https://mizugadro.mydns.jp/t/index.php/Superfunctions_in_Russian}\\
\url{https//www.ils.uec.ac.jp/~dima/BOOK/202.pdf}\\
\url{https://mizugadro.mydns.jp/BOOK/202.pdf} Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014; In Russian. 
Sorry, the Russian characters are not copypasted well}.
I did great efforts translating the Russian expressions into English; figures \ref{oblako},\ref{conductor}  prove this. 
\vskip 3pt

I provide references to the English translations of the classical books cited in the Russian version.
However, in many cases, the translation is not possible. Often, it is difficult to find the English equivalents of the references cited. So, I give up trying to translate the Book; %, my skills are not sufficient to do it well. I suspect, nobody can do it well; so,
I re-write it in English. 
\vskip 6pt

I reuse some formulas and figures, that are available at the free sites; in particular, at Mizugadro and Citizendium. 
The links are mentioned in the Supplement \ref{SupplementSites}, at page \pageref{SupplementSites} . I load gallery of the images at \url{https://mizugadro.mydns.jp/t/index.php/Category:Book} ; I hope, this server will last longer, than the previous holder of TORI at
https://tori.ils.uec.ac.jp/TORI, that had been attacked 2013.02.27; then, the access 
had been disabled.
It took certain efforts to arrange the clone at \url{https://mizugadro.mydns.jp/t}. 
The reason of the aggression could be the critical article prepared for ``Physics Today''
\footnote{\url{https://mizugadro.mydns.jp/PAPERS/2013physToday.pdf} D.Kouznetsov. Corruption in Russian science. 2013, preprint}.
Unfortunately, namely this way the politicians answer the critics, while they cannot build any logical argumentation.
I hope, those politicians pay to administration of our university some significant amount per each day, while TORI is not available at its original URL.
I have no other explanation of the attitude or our administration, who had post-factum approved that barbarian attack.
I try to make Science, not the criminal investigation, so, I mention only the scientific point of view on the events. However, I hope, one day, the professional criminalists carefully investigate the case, as well as the origins of other attacks happened
2013.02.27,
2014.02.27,
2015.02.27,
2016.02.27,
2017.02.27,
2017.03.04 (lated few days since the expected date 2018.02.27).
\footnote{%Periodic sequence of brutal attacks is observed with period one year.\\
\url{https://mizugadro.mydns.jp/t/index.php/Tori_attacked} \\
\sx{.96}{\url{https://budclub.ru/k/kuznecow_d_j/toriattacked.shtml} D.Kouznetsov. TORI attacked, 2013}}%
\vskip 2mm

I allow the free use of my results, and I try to simplify the use. In particular, I provide generators of images, used in the Book.
I would not specify this in details, but one recent observation indicates that the problem is very serious.
Since year 2014, the strange phenomenon is observed in the resources of wikimedia:
the administrators claim, that publication of my results violate my authors rights. The same, and even worse, happen with results by other authors from the USSR and, especially, texts of the Soviet dissidents. The administrators ignore the permissions provided by the copyright owners, and remove the texts and the images, claiming, that the loading or the texts and images are 
``Copyright violation''. The similar phenomenon could take place in the Nazi Germany. The nazi used to arrest, to kill the opponents, and withdrawn their books from shops and libraries with pretext of defence of the intellectual property.
They killed many authors, and, if the author is already dead, the fascists claim, that now the State becomes owner of the copyright property. However, the State, after to kill the authors, prohibit the distribution of their works.
May be, the professional historian corrects me, that the German nazi acted in a little bit different way. 
But everyone may look at wikisource, and, especially, to its Russian version 
and see, that namely this happens at wikimedia, and, especially, since year 2014. Perhaps, the Russian Soviet veterans decided, that the invasion into Ukraine and  plundering of Crimea is not sufficient, and begun to plunder also the wiki resources.
I mention this in the special statement
\footnote{
\url{https://mizugadro.mydns.jp/t/index.php/Kouznetsov,_permission}
Open letter by Dmitrii Kouznetsov about massive removal of texts and images from Wikimedia projects with pretext of defence to the copyright of the authors. Wed, Mar 18, 2015 at 11:23}. 
\vskip 2mm

I type the paragraph above in order to stress, that I had issued the permission to distribute my results under the free GNU license;
the only attribution is required. The attempt to ``defend'' my author's right in the way described above, is a fraud: several years ago I had allowed the free distribution, and I provide the links, that confirm this. In particular, the permission refers to the plots, used in this book and their generators. I specify the URL of the image at each picture. If some administrator claims, that use of this image under the "free" license violates my copyright, let me know. I think, such an administrator should be qualified as liar, impostor, knave, thrive; and his/her/its attempts to defend my author's right in such a way should be qualified as a fraud.
In other words, use the images from this Book, as well as from Russian version, for free, but do not forget to attribute the source.

\section{Physical mathematics}
\baselineskip 19.8pt

While writing this Book, I was asked about its genre. I searched the answer in various databases, and it seems, the closest qualification is ``Physical mathematics'' \cite{fizmat,physmat,euclid}. 
\vskip 9pt

Term ``physical mathematics'' should not be confused with ``mathematical physics''. The order of words is important. 
Similar examples exist in the Quantum Mechanics. Let $P$ be momentum of a particle, and $Q$ be its coordinate; then 
$PQ\!-\!QP\!=\!-\mathrm i \hbar$.
\vskip 4pt

One student could not catch the sense of non-communing operators. She tried to understand, why operator of creation does not commute with operator of annihilation. The Professor tried different explanations, then he found the strong example for female students; he asked:\\
{\em Could you explain me, please, why operation  {\bf TO CLEAN-UP} does not commute with operation  {\bf TO DIRT-DOWN}  ?} \\
The student was content, she said:\\
{\em Thank you, Professor! This is very good example! Now I understand all the Quantum Mechanics!}\\
Professor also got a good lesson; he realised, how to teach Quantum Mechanics, keeping women and female logic in mind
\footnote{\url{https://pda.anekdot.ru/id/658766}  Bogorad Victor. Мысли, Мысли. (2013, In Russian) \\
\url{https://mizugadro.mydns.jp/t/index.php/Female_logic} D.Beklemishev. Female logic. (2013)\\
\url{https://www.ams.org/notices/201309/rnoti-p1156.pdf} 
J.M.Deshler, E.A.Burroughs. Teaching Mathematics with Women in Mind. Notices of the AMS, V.60, No.9, p.1156-1163.
}.
\vskip 9pt

Term ``mathematical physics'' denotes the mathematical exercises with equations that come from Physics.\\
Term ``physical mathematics'' means, that the common scientific tools are used to check mathematical conjectures; the rigorous mathematical deduction appears as a possible scientific method, but not an imperative.

\section{TORI axioms}
\baselineskip 19.9pt

In this Book, I use the TORI Axioms. % I overview these axioms in this section. 
This section retells some results from publications in ``Uspekhi''
\cite{ufn} and J. of Modern Physics \cite{2013jmp}. These results refer to definition of term ``Science''. TORI Axioms appear as parts of the definition of this term.
\vskip 9pt

{\bf Science is kind of knowledge, activity and research, based on concepts, that have all the six properties below:}
\vskip 11pt


{\bf 1. Applicability:} Each concept has the limited range of validity, distinguishable from the empty set.
\vskip 9pt

{\bf 2. Verifiability:} In the terms of the already accepted concepts, some specific experiment with some specific result, that confirms the concept, can be described.
\vskip 9pt

{\bf 3. Refutability:} In the terms of the concept, some specific experiment with some specific result, that negates the concept, can be described.
\vskip 9pt

{\bf 4. Self-consistency:} No internal contradictions of the concept are known.
\vskip 9pt

{\bf 5. Principle of correspondence:} It the range of validity of a new concept intersects the range of validity of another already accepted concept, then, the new concept either reproduces the results of the old concept, or indicates the way to refute it. (For example, the estimate of the range of validity of the old concept may be wrong.)
\vskip 9pt

{\bf 6. Pluralism:} Mutually-contradictive concepts may coexist; if two concepts, satisfying requirements  1-5 above have some common range of validity, then, in this range, the simplest of them has priority.
\vskip 16pt

The 6 properties, requirements above are {\bf TORI axioms.}
They are specified also at %These axioms are used as basis of site
\url{https://mizugadro.mydns.jp/t}.
\vskip 9pt

The main results, that had been presented at TORI and related to this Book, are published in Scientific Journals \cite{2013jmp,2013or,sutra,susin}.
In order to simplify the referencing, the axioms above needed a special name. Having poor fantasy, I just keep the initial abbreviation, TORI, that means Tools for Outstanding Research and Investigation. The Russian transliteration ``ТОРИ''  has similar meaning (``Теоретические Основы Революционных Исследований'').
Many results, reported at the conferences and in the scientific journal, look so similar, that it is difficult to find, what 
namely new and interesting does each of them suggest. They barely satisfy the TORI axioms, if at all. The non-trivial results appear to be ``outstanding'', ``revolutionary'', in the sense, that they stay a little bit out of the main trend of scientific research. 
\vskip 9pt

Articles of  site 
\url{https://mizugadro.mydins.jp/t} collect tools: concepts, formulas, algorithms, pictures, that I consider to be ``outstanding'' and useful in the scientific research. I assume, that the ``instanding'' researches are already well represented in Citizendium, Wikipedia and scientific journals, and there is no need to repeat them again. In particular, I load to TORI generators of figures, used in this book.
The idea is, that each code can be downloaded by the Reader, compiled, executed, and, if necessary, modified for the new 
(and, perhaps, unexpected for me) applications.
\vskip 9pt

\begin{wrapfigure}{r}{178pt}
\vskip -4pt
\sx{.4}{\begin{picture}(300,390)
\put(8,0){\ing{uro2}}
\put(0,0){\color{white}\rule{400pt}{40pt}}
\end{picture}}
\vskip -32pt
\caption{\large Server felt down \label{uronila}}
\vskip -8pt
\end{wrapfigure}
I load the generators also to other sites, in order not to make an impression, that the politicians may completely block the access to my results
attacking another server. In this sense, {\em Manuscripts don't burn} \cite{bulga}.

In the Supplement, I suggest the links to other sites for the case if some server is not reachable.
The same, for the case, if it suddenly falls down, as it is shown in 
Figure \ref{uronila}.
\vskip 9pt

With links, supplied to figures, the Reader can use and develop the formalism of superfunctions just from the place, until I have advanced.
Just pick-up the figure you like and load its generator. 
\vskip 9pt

\section{What for this book is}
\baselineskip 19pt

Since publication in 2009 of the article about holomorphic tetration
\cite{analuxp}, 
I expected, the mathematicians can do the rest of the job by themselves. 
That publication refers to the special case of equation \rf{prologeq}, with $T\!=\!\exp$; and the natural tetration as solution is considered, $F\!=\!\mathrm{tet}$.
\vskip 9pt

Seven years passed by since \cite{analuxp}, but superfunctions did not appear in descriptions of the algorithmic languages. Neither tetration, nor superfactorial \cite{factorial}, nor other superfunctions I had reported appear as built-in functions. Until now, the function Nest \cite{nest} in language ``Mathematica'' requires, that the number of iterate is integer.
If one approximates some bell-like function, then, usually, the fantasy of colleagues does not go farther than the Gaussian exponential, hyperbolic secant or Lorentian. One could use also, for example, the half-iterate of the exponential \cite{kneser}, id est, such function $f$, that $f(f(z))\!=\!\exp(z)$, instead of the exponent in $2/(\mathrm e^x\!+\!e^{-x})$, as well as other non-integer iterates of various functions, but one needs the formalism of superfunctions for the evaluation. This Book describes such a formalism.
\vskip 9pt

Most of the Mathematical Analysis in schools and universities is built-up on the first three ackermanns: addition, multiplication, exponentiation and their inverse functions in various combinations.
This arsenal could be greatly extended, including, for example, the 4th ackermann, which is tetration, or the 5th ackermann, that is pentation. However, one needs formalism of superfunctions to evaluate tetration and pentaiton \cite{vladie,pentation}
\vskip 6pt

When physicists analyse the nonlinear response of a medium, they try to make the sample optically-thin; 
otherwise, the re-absorption and 
saturation lead to mistakes \cite{perpe1,perpe2,perpe3,perpe4,perpe5,perpe6}. 
For the thin sample, the precision of measurements is low. Superfunctions allow to work with thick samples, and still reconstruct behaviour of intensity inside the sample \cite{thailand,2013or}. 
\vskip 6pt

Many researchers believe that namely their work is especially important. I am not an exception. From my point of view, tetration and arctetration, as well as other superfunctions and abelfunctions, will be important for science since century 22; so important, as the exponential and logarithm are important since century 19. 
This Book explains, how to calculate superfunctions and what can one do with them.
\vskip 6pt

When the fundamentals of superfunstions had been formulated, one of my coauthors, Akira Shirakava 
\cite{akira0} had declared, that it is not possible, to recover distribution of intensity of light in the amplifier from its transfer function.
\cite{akira1}. Then, the algorithm for this recovery had been presented
\cite{sqrt2,thailand,2013or}, and 
Shirakava said, that the formalism of superfunctions is too complicated
\cite{akira2}. 
Through all the mutual misunderstandings, I consider Shirakawa as great specialist in fibre lasers, 
and I take into account his complains.
I try to make this book popular. So, I cite not only scientific articles.
Also, I try to avoid complicated mathematics, at least in some figures.
I still suppose, that the Reader can distinguish an Integral from a Logarithm, and has some idea, what is square root of minus unity. 
If the Reader, in addition, had seen any Cauchy integrals, this is a good advantage.
\vskip 3mm
In order to simplify the reading, I try to avoid long sentences and digressions, although not always it is possible. 
In order to explain this, I suggest the example:
\vskip 3mm
\baselineskip 14.6pt

Happy is the traveller who, after a long and heavy journey, with its cold and humidity of air-conditioners, with long waiting-lines at the passport control, curious to under ware the special control officers, delayed flights, lost luggage, going to the opposite direction taxi drivers, sees at last the familiar roof with its lights approaching to meet him. And there rise before his mind the familiar rooms, the delighted outcry of the servants running out to meet him, the noise and racing footsteps of his children, and the soothing gentle words interspersed with passionate kisses that are able to efface everything gloomy from the memory. Happy the man with a family and nook of his own, but woe to the bachelor! \\
Happy is the researcher who, passing by the strange results, paradoxes and their  wrong, inconsistent interpretations, attaches himself to phenomena that display the loftiest virtues of scientific achievements, who, from the great whirlpool of figures flitting by him daily, has selected only the few exceptions, for which the answers are already known;
who has never tuned his research to a less exalted key, has never stooped from his 
pinnacle to really new and unexpected phenomena. His fair portion is doubly worthy of envy; he lives in the midst of them as in the midst of his own family  and, at the same time, his fame resounds far and wide. He clouds men's vision with enchanting incense; he flatters them marvellously, covering up the gloomy side of science and life, and exhibiting to them the noble man. All run after him, clapping their hands and eagerly following his triumphal chariot. \\
They call him a great world-famed scientist, soaring high above every other genius as the eagle soars above the other birds of the air. Young ardent hearts are thrilled at his very name; responsive tears gleam in every eye. . . . No one is his equal in power — he is a God!
But quite other is the portion, and very different is the destiny of the writer who sees and reveal the phenomena strange, that are out of the commonly accepted theories and contradict to the obvious commonly-accepted common sense, calling for the revision or at least some critical analysis of the widely recognised results, that already have assigned the highly prestigious awards and huge grants and foundations..
En fin, he'll not escape from the scientific council, who keep the old paradigms and any doubt consider as a sin, as a kind of heresy..
Without compassion, such a researcher is left by the roadside like the traveller without a family. Hard is his lot and bitterly he feels his loneliness.
\JP{But quite other is the portion, and very different is the destiny of the writer who has the temerity to bring to the surface what is ever before men's sight and  is unseen by their indifferent eyes — all the terrible overwhelming mire of triviahties in which our life is entangled, all that is hidden in the often cold, petty everyday characters with which our bitter and dreary path through life swarms, and with the strong hand of a relentless sculptor dares to present them bold and distinct to the gaze of all ! It is not for him to receive the applause of the people, it is not his lot to behold the grateful tears and single-hearted rapture of souls stirred by his words ; no girl of sixteen, with her head turned, flies to meet him with heroic enthusiasm. It is not his lot to lose himself in the sweet enchantment of sounds he has himself evoked. And lastly, it is not his lot to escape the contemporary critic, the hypocritical callous contemporary critic, who will call his cherished creations mean and insignificant, will assign him an ignoble place in the ranks of writers who have insulted humanity, will attribute to him the qualities of his heroes, will strip him of heart and soul and the divine fire of talent. For the contemporary critic does not recognise that the telescope through which we behold the sun and the microscope which unfolds to us the movements of unnoticed insects are equally marvellous. For the contemporary critic does not recognise that great spiritual depth is needed to light up a picture of ignoble life and transform it into a gem of creative art. For the contem-porary critic does not admit that the laughter of lofty delight is worthy to stand beside exalted lyrical emotion, and that there is all the world of difference between it and the antics of clowns at a fair ! All this the critic of to-day does not admit, and he will turn it all into the censure and dishonour of the unrecognised writer. Without sympathy, without response, without compassion, he is left by the roadside like the traveller without a family. Hard is his lot and bitterly he feels his loneliness. ..}
\cite{gogol}
\vskip 11pt

\baselineskip 18.8pt
I promise, that the above is longest digression in this Book. Trying to keep the Book thin, the philosophic exercises about the place of Science in the Human knowledge
\cite{ufn,mesto} 
are published separately. It is not possible to cover Everything (id est, the Full Set). 
Since the beginning, I try to specify, what is in the book, and what is absent in the Book.

\section{What is present in the Book}
\baselineskip 18.8pt

This Book suggests the general approach for construction of superfunctions and various algorithms of their evaluation.
Superfunction is solution $F$ of equation \rf{prologeq}; I repeat it here: $F(z\!+\!1)\!=\!T(F(z))$. This equation is called ``Transfer equation'', and the given function $T$ is called ``transfer function''.
\vskip 9pt

In this book, I describe various algorithms for evaluation of superfunctions. The choice of the algorithm depends on the %stationary points, i.e., 
fixed points of the transfer function, which are solutions $L$ of equation $T(L)=L$; and, also, depends on the additional requirements: what properties do we expect from the required solution $F$ of equation  \rf{prologeq}. These properties, being postulated, should provide the uniqueness of the solution, but still should allow its existence and should indicate some way(s) of the evaluation.
\vskip 9pt

The inverse of superfunction $F$ is called here ``abelfunction'' $G\!=\!F^{-1}$. The abelfunction satisfies the Abel equation $G(T(z))=G(z)\!+\!1$.
In this book, I consider many examples of the transfer function $T$, superfunction $F$ and the abelfunction $G$. 
I suggest ways of the efficient evaluation. Efficient means, that the functions (and corresponding figures) are computed in the real time and with good precision. With use of the complex double arithmetics, the superfunctions and abelfunctions are evaluated with 14 decimal digits.
I load the figures together with their generators to  
\url{https://mizugadro.mydns.jp/t/index.php/Category:Book}; 
the readers can check them with C++ and Latex.
\vskip 9pt

By default, I assume, that the parameters have real values, while the arguments of the functions may have complex values.
For the illustrations, the complex maps are essential. In particular, the complex map of the natural tetration is used at the front cover of this Book.
\vskip 8pt

I am more physicist, than mathematician. For physical applications, the real values of the argument are of special interest.
Usually, the real values allow the straightforward interpretation and, sometimes, the application.
For this reason, I supply also the graphics for the real values of the argument.

\section{What is absent in the Book}
\baselineskip 18pt

In this book, there are no accurate specifications, which values are allowed for parameters and arguments of the functions considered. I specify them only in the cases, when they are essential for the consideration. For each expression, the readers are supposed to examine the range of applicability as the exercise. The reader is not a vessel  to be filled, but a candle to be lighted
\footnote{\url{https://quoteinvestigator.com/2013/03/28/mind-fire/} \baselineskip 9pt
The Mind Is Not a Vessel That Needs Filling, But Wood That Needs Igniting. March 28, 2013. .. None of the examples came with citations:\\
Education is the kindling of a flame, not the filling of a vessel —Socrates\\
Education is not the filling of a pail, but the lighting of a fire. —William Butler Yeats\\
Education is not the filling of a pail, but the lighting of a fire. —Plutarch\\
The mind is not a vessel that needs filling, but wood that needs igniting. —Plutarch ..
}.
\vskip 6pt

I try to define functions in such a way, that at least some vicinity of the positive part of the real axis belongs to the range of holomorphism;
although, not always this is possible. This book %(At least its Russian version) 
seems to be the first monograph with description of efficient methods of evaluation of superfunctions and abelfunctions, and this should be considered as an excuse for the heuristic style of presentation of the results.
\vskip 6pt
For the same reason, in this Book, there are no rigorous proofs of convergence, existence and uniqueness of solutions.
In particular, the deductions, elaborated in collaboration with Henryk Trappmann, are not included here.
The Readers who like the deep drilling
\footnote{Importance of the deep drilling can be illustrated with the limerick below: \baselineskip 9pt\\
One Chinese colleague called Lee \\
Was drilling his girl at the sea;\\
She told him: {\em Stop plumbing,\\
Somebody is coming!}\\
And Lee had replied:{\em That's me!}
}
are invited to download our original articles
\cite{sqrt2,bitbucket,uniabel,e1e,log}, plumb through the deduction expanded with the Saxon style of pedantic Henryk, drill the examples and ask questions.
Indications of the errors
are of especial interest.
\vskip 2mm

From my point of view, the theory of superfunctions is 
new branch of the Mathematical Analysis, and the serious drilling and plumbing require efforts of some institute with several research groups in order to bring it to the beautiful and rigorous shape, like the theory of the differential and Integral calculus and other mathematical analysis.
Here, I deal with functions of single variable; the generalisation to multidimentional case is not presented.

\vskip 2mm

History to development of theory of superfunctions and abelfunctions begins approximately since century 19, and can be also interesting, but it falls out of the topic of this Book.
The overviews of the pre-historic publications can be found in the Internet with the following keywords: {\bf 
Abel function, 
iterated functions,
tetration, 
superexponential, 
super\-function}.
Some of them can be considered as scientific, according to the definition above; I mean, the TORI axioms. This Book is dedicated to the Superfunctions, not to TORI axioms, so, these axioms are presented here only schematically; however, the more detailed description is available \cite{ufn,mesto,science,sciencer,Hayka}.
\vskip 2mm

In this Book, there is no detailed discussion (and sometimes nor even citation) of publications of colleagues, who tried to evaluate the superfunctions during the last couple of centuries. In order to see, what had been done in the past centuries, one can look at the publications by colleagues: 
Neils Henryk Abel \cite{abel},
Peter Fatou \cite{fatou}, 
Hellmuth Kneser \cite{kneser},
Jean Ecalle \cite{ecale}, 
G.Szekeres \cite{szekeres},
Peter Walker \cite{walker};
more references are suggested in year 2014 by 
J.Bonet and P.Domanski \cite{bonet}.
In addition, some links are listed in the Supplement (page \pageref{SupplementSites}).
\vskip 2mm

\begin{wrapfigure}{r}{250px}
%\sx{.333}{\begin{picture}(600,580)
%\put(20,10){\ing{720px-Flammarion.jpg}}
\sx{.82}{\begin{picture}(220,220)
\put(10,0){\ing{FlammarionC288}}
\end{picture}}
\vskip -10pt
\sx{.64}{\url{https://commons.wikimedia.org/wiki/File:Flammarion-color.png}}
\vskip -8pt
\sx{.98}{\url{https://en.wikipedia.org/wiki/Flat_Earth}}
\vskip -3pt
\caption{%Flat 
Earth \label{FlatEarth} in the old model of the Universe}
\vskip -6pt
\end{wrapfigure}
I see %the 
analogy with 
manuals on Geography and Astronomy, that have no need to begin with the historic excursus, describing the 
Flat Earth (Figure \ref{FlatEarth}), %\url{https://en.wikipedia.org/wiki/Flat_Earth},
nor the Geocentric model by Ptolemy.
Perhaps, some Readers may consider such an analogy to be too pretentious and ambitious;
but the methods reported before century 21 are, indeed, non-efficient; so inefficient, that
nobody could plot a complex map of any non-trivial superfunction with methods reported in century 20.
Henryk Trappmann attributes this to my routine \url{https://mizugadro.mydns.jp/t/index.php/Conto.cin}, which is now available also at various wikis.
I think, the matter is not my plotter, but the beautiful and efficient representations of functions, that allow the quick and precise evaluation. 
In principle, these algorithms can be programmed at Mathematica, even at Maple, and still generate the same figures; although the efficiency of Maple (and, especially, Maple-10) in generating of figures allows to use that software as an illustration to the known proverb ``to press a key, to have a tea'', as it is described in the Supplement, page \pageref{MapleAndTea}.
I do not consider that poem as a part of this Book, but the poem and figure \ref{figmaple} explains, why the most of figures in this Book are generated as the direct EPS graphics from the C++ programs.
\vskip 2mm

I do not describe in this book algorithms, suggested after year 2015, 
for example, those by William Paulsen and Samuel Cowgill \cite{paulsen2016,paulsen,paulsencowgill,cowgill,paulsen2019}.

\vskip 2mm
I do not include into this Book the codes of the programs (figure \ref{figmaple} is an exception), used for evaluation of functions and plotting graphics. The algorithms are simple; any student, after a semester of programming in any language, that supports the complex arithmetics, can easy program the same, following the description from this Book.
At figures, I indicate the URLs,  where the code is loaded.
\vskip 2mm

Some results of this book can be reproduced with the Schr\"oder equation and the Schr\"oder functions.
In this Book, there is no detailed description of properties of the Schr\"oder equation, nor analysis of the Schr\"oder functions; these functions are only mentioned.
I could not construct any example, that can be solved with  Schr\"oder functions, but cannot be solved with superfunctions.
Some properties of the  Schr\"oder equation can be found at\\
\url{https://mathworld.wolfram.com/SchroedersEquation.html}  and\\
\url{https://en.wikipedia.org/wiki/Schroeder_equation}
\vskip 2mm

Preparing the results for the Book, I tried to refute each of the conjectures suggested. In principle, it is possible.
It is sufficient to construct any example of a transfer function $T$ such that the superfunction $F$, constructed by the methods, described in the Book, does not satisfy the transfer equation \rf{prologeq}.
Or to construct two different superfunctions, that satisfy the additional conditions, that are declared to provide the uniqueness.
My attempts to reject my claims failed. I could not find in the literature any rejecting example, nor I could construct it by myself. Only for this reason, the rejecting, refuting examples are not included in the Book. 
\vskip 2mm

Many colleagues, as myself (since the childhood), prefer to watch pictures in a book, rather than to read it. Therefore, I try to illustrate each page with at least one figure. While, this goal is not yet achieved, but I hope to approach it closer in the future editions. 
\vskip 2mm


\chapter{~ Introduction \label{intro}}
\baselineskip 18.4pt

\begin{wrapfigure}{r}{160px}
\sx{.32}{\begin{picture}(160,2)
\put(-180,40){\ing{snowball}}
\put(-100,384){\color{white}\rule{200pt}{40pt}}
\put(-100,390){\sx{2.4}{$F(0)$}}
\put(290,152){\color{white}\rule{230pt}{60pt}}
\put(290,160){\sx{2.4}{$T(F(0))\!=\!F(1)$}}
\end{picture}}
\vskip -20pt
\caption{Snowball of mass $F(0)$ gets mass $T(F(0))\!=\!F(1)$ after to roll once from the hill. \label{snow}}
\end{wrapfigure}
Iterates and Superfunctions arise naturally at description of a sequence of similar transformations of any quantity in a homogeneous physical system. This transform may refer to accumulation of mass by a snowball, that rolls down the hill, covered with sticky snow (Fig. \ref{snow}).
This may be attenuation of a shock wave in one of the successive sections of automobile muffler. 
The transform may  refer to change of intensity of light, passing one section of a laser amplifier or a saturable absorber. 
I hope, the Reader can suggest more examples of this kind.
\vskip 2mm

Let the state of the system before the transformation be described with parameter $x$, and, after transformation - with parameter $y$;
and let these two quantities be related with $y\!=\!T(x)$. Then, function $T$ is called ``transfer function''. It describes the transformation of the signal in the system. 
\vskip 2mm

System, that performs the transformation, can be called ``filter''. Transformation of signal $x$ in a filter with transfer function $T$ can illustrated with expression
\be
x \rightarrow \framebox{filter} \rightarrow y\!=T(x)
\eL{frame1}
 Output of one filter can be directed to the input of another identical filter. This can be expressed with
 \be
x \rightarrow \framebox{filter} \rightarrow T(x) \rightarrow  \framebox{filter} \rightarrow T(T(x))
\eL{frame2}
One can write the similar expressions for combination of 3 filters and so on. In these notations, the number of filters combined is supposed to be a positive integer. The main idea of this Book is, that the number of filters has no need to be integer. In this chapter, the notations are suggested to 
describe this.

\section{Use of superscript}
\baselineskip 19pt
From the point of view of physics, every iterate describes some combination of identical filters.
Let each filter be characterised with the transfer function $T$.
If we have signal $z$ at the input, then, at the output, the signal is $T(z)$.
\vskip 2mm

As it is mentioned in the Preamble of this chapter, the output of a filter can be directed to another similar filter, and the transfer function of the resulting combination can be expressed with
\be
T(T(z))=T^2(z)
\eL{IntroT2}
In the right hand side of equation \rf{IntroT2}, the notation with superscript is used. I am not first to suggest this notation;
Walter Bergweiler had used such a notation in the past century
\cite{walter}. 
If there is superscript at name of some function, and the expression in this superscript can be interpreted as a number, then it 
is number of the iterate.
The zeroth iterate of a function is supposed to be identical function (its value is the same as value of its argument):
\be
T^0(z)=z
\ee
First iterate of some function $T$ is the same function $T$, and the minus first iteration corresponds to the inverse function.
For example, at the sinusoidal transfer function
$T\!=\!\sin$, we have
\be
\sin^{-1}(z)&=&\arcsin(z) \\
\sin^{0}(z)&=&z\\
\sin^{1}(z)&=&\sin(z)\\
\sin^{2}(z)&=&\sin(\sin(z))\\
\sin^{3}(z)&=&\sin(\sin(\sin(z)))
\ee
and so on.
In some textbooks, the notation is used (but not declared), when expression $\sin^a(z)$ has sense of $\sin(z)^a$. 
Such a notation leads to confusions: at $a\!=\!-1$, expression $\sin^a(z)$ might mean   $\ds \frac{1}{\sin(z)}$; but, according to another commonly used notations for the inverse function, it should mean  $\sin^{-1}(z)\!=\!\arcsin(z)$.
\vskip 2mm

In such a way, in this Book, the expression in the upper superscript at the name of a function is interpreted as number of iterate. However, if the superscript is just ``prime'', almost vertical stick, it indicates the derivative; for example, $\sin'\!=\!\cos$; it is also usual notation. 

\section{Transfer equation}
\baselineskip 19.8pt
The use of the superscript allows to illustrate expressions \rf{frame1} and \rf{frame2} 
with figure \ref{figfilters}. 
The figure shows the combination of identical filters with transfer function $T$ each. 
I assume $T$ to be some given, known holomorphic function.
The range of holomorphism is supposed to be wide enough, to cover the needs  of the following consideration.
\vskip 2mm
\begin{figure}[t]
\sx{1.02}{\begin{picture}(200,30)
\put(6,22){$f_0$}
\put(106,22){$T(f_0)$}
\put(206,22){$T^2(f_0)$}
\put(306,22){$T^3(f_0)$}
\multiput(10,0)(100,0){3}{\framebox(100,16){filter}}
\put(380,0){$\longrightarrow$}
\put(390,-9){$z$}
\put(0,-12){$F(0)$}
\put(100,-12){$F(1)$}
\put(50,-22){$F(z)$}
\put(200,-12){$F(2)$}
\put(148,-22){$F(z\!+\!1)$}
\put(300,-12){$F(3)$}
\end{picture}}
\vskip 18pt
\caption{Combination of filters, each of them has transfer function $T$ \label{figfilters}}
\vskip 18pt
\end{figure}

At the left hand side of figure \ref{figfilters}, in at the entry to the first filter, let the signal is characterised with some fixed value 
$f_0$. Then, after to pass the filter, it becomes 
$T(f_0)$; after to pass the second filter, it becomes 
$T^2(f_0)\!=\!T(T(f_0))$, and so on. These quantities are specified above the vertical bars, that 
mark the end on one filter and beginning of the next filter. 
\vskip 6pt

The notation can be even shortened, to count number of passes of the filter. 
Let $f_0$ be $F(0)$; 
let $T(f_0)$ be $F(1)$; 
let  $T^2(f_0)$ be $F(2)$, and so on.
In such a way, function $F$ of non-negative integer argument can be defined.
\vskip 4pt

Now I need the strong assumption. Let the filters, inside, are uniform, and act as some kind of continuous homogeneous nonlinear medium, that transfers the signal by some fixed (although, may be, not yet known) way; but,
after to pass each section of the combined filter, 
the signal is transformed 
with known transfer function $T$.
\vskip 6pt


We may consider coordinate $z$ along this combined filter, and treat it as
a continuous medium.
It is convenient to choose the length of a single filter as a unit to measure the coordinate along the combined amplifier. Then, $F(z)$ may have sense of signal at coordinate $z$; for integer values of $z$, values of function $F(z)$ are already known. The question is, how to define, determine, evaluate function $F$ for non-integer values of $z$.
\vskip 2mm

If for some non-integer position $z$, for example, between zero and unity, the signal is $F(z)$, then, in the uniform medium, after a pass length unity along the composed filter, the signal should be transformed with the same transfer function $T$.
This can be expressed with the transfer equation \rf{prologeq} mentioned in the Prologue. 
I repeat this equation:
\be
F(z\!+\!1)=T(F(z))
\eL{introeq}
The transfer equation \rf{introeq} specifies, how does the value of function $F$ changes, while its argument gets the unity increment.
In such a way, figure \ref{figfilters} indicates the physical meaning of the main equation in this Book; the figure suggests the physical 
implementation of the transfer equation \rf{introeq}.
\vskip 2mm

\section{Values of the argument}

In this Book, I assume, that the signal, transferred through the filter, is characterised 
by a single number, real or complex.
In principle, term ``transfer function'' allows generalisation to the multidimensional case; then the transfer function appears as a functional, and its argument 
may have sense of a vector or a function. 
As it is mentioned in the Preface, here I consider only the case of a single variable: 
it happened, that even the single-dimensional case causes a lot of confusions at the internet forums and discussions.
It is methodically-incorrect, to develop the multidimensional generalisations, while the colleagues have doubts even in the single-dimensional case.
According to the definition of science \cite{ufn,science} in the Prologue, first, the results bout the single-dimensional case should be presented, that still refute the commonly accepted point of view, that the recovery of the signal inside a homogeneous system from its transfer function is not possible
\cite{yoneda,kaminski,akira1,akira2}.
\vskip 2mm

One of objections refers to the multitude of solutions. Consideration of complex values of argument of the transfer function allows to reduce the class of possible solutions
(and sometimes even provides the uniqueness of the solution). 
In such a way, consideration of complex argument is more important, than analysis of some multidimensional signal.
For this reason, in this Book, I assume, that the argument of the transfer function is a complex number.
Also, I assume, that the transfer function is holomorphic, at least at some vicinity of some part of the real axis.
\vskip 2mm

\section{Nest}
\baselineskip 19pt

\begin{wrapfigure}{r}{90px}
\sx{.45}{\begin{picture}(200,36)
\put(-30,-28){\ing{nestbird1}}
\end{picture}}
\caption{Nest \label{nestbird}}
\vskip -12pt
\end{wrapfigure}


For iterates of functions, in algorithmic language Mathematica, there is special routine 
{\bf Nest}.
It requires 3 arguments.
The first one indicates the name of function that 
should
be iterated.
The second argument indicates the initial value at the iterates.
The third (and last) argument specifies the number of iterate.
The call looks as follows:
\be
\mathrm{Nest}[ f , x , z ]
\label{matnest}
\ee 

In Mathematica, the arguments of a function appear in squared parenthesis; and expression \rf{matnest}
means 
$f^z(x)$. (However, term ``Nest'' may have also different meaning, as it is shown in figure \ref{nestbird}.)
\vskip 2mm

At least until year 2017, the implementation of Nest 
requires, that 
the last argument can be simplified to a positive integer constant. Even 0th and minus first iterates are prohibited. 
Perhaps, the designer assumed, that only the integer number of iterate may have sense and meaning.
\vskip 2mm

Actually, the non-integer iterates do have sense, as it is mentioned in the preamble of this chapter.
One more example is considered in the next section.
\vskip 2mm


\section{Fibre amplifier}

This section describes the example, that shows the sense of the non-integer iterates.
This example refers to the fibre amplifier. I type ``fibre'' in order to indicate, that the signal is confined in two directions, and only the power in the fibre as function of the coordinate along it is subject of consideration.
A lot of physical effects are dropped out in this consideration: change of the spectral content of the signal, self-peaking, spontaneous emission, etc.. So, I consider here the simplest case, but for this model I want to get the exact solution.
\vskip 2mm

Assume, some Manufacturer gives to some Physicist a piece of one meter long of the optical fibre amplifier, together with the system of pumping, and asks the Physicist to investigate, how the power of light inside grows during the amplification, but Manufacturer does not allow Physicist to cut the fibre to measure, what is inside. Physicist knows the only, that the fibre is uniformly pumped. We may assume, that some system of lateral delivery of pump is used \cite{slab1,slab2, slab3, fidi2}. 
\vskip 2mm

Physicist can measure the transfer function $T$ of this fibre. With this transfer function, Physicist can say, what should be the transfer function of similar piece of fibre of length 2 meter. Assuming, that the source of pump is delivered together with each piece of the fibre, the transfer function of the piece of two meter is $T^2$; that of the 3 meter piece should be $T^3$ and so on; if $z$ pieces of the fibre are combined, then, the transfer function is $T^z$.
\vskip 2mm

Iterates $T^z$ are pretty clear, while $z$ is integer. But how about non-integer values? For example, what is transfer function to the piece of half meter long?  
\vskip 2mm

As it is mentioned in the previous section, in Mathematica, the transfer function of piece of $z$ meter length could be expressed with
$T^z(x)=\mathrm{Nest}[T,x,z]$. 
Unfortunately, such an expression is not yet interpreted correctly at non-integer $z$.
\vskip 2mm

How to express the non-integer iteration of a given transfer function?
These question can be analyzed with the transfer equation \rf{introeq}.
The preliminary analysis is suggested in the next section.

\section{Transfer equation and the Abel equation}

Through this book, the transfer equation \rf{prologeq},\rf{introeq} is repeated (and used) again and again, and, in particular, here:
\be
F(z\!+\!1)=T\big(F(z)\big)
\eL{transfereq}
I remind, that $T$ denotes the transfer function,
$z$ may have sense of coordinate along direction of propagation of some signal, and
function $F$ expresses dependence of the signal on this coordinate. (However, coordinate  $z$ may have also any other meaning.)
\vskip 6pt

In Laser Science, term ``signal'' denotes the wave (light), that is amplified; even if no information is transferred with this light.
In this book, the length is measured in units of the length of the amplifier. 
The generalisation for the arbitrary units is straight-forward.
\vskip 6pt

For given transfer function $T$, solution $F$ of the transfer equation \rf{transfereq} is called ``superfunction''. The inverse function
$G\!=\!F^{-1}$ is called ``Abel function'' or ``abelfunction'' for the same transfer function $T$.
The abelfunction $G$ satisfies the Abel equation
\be
G(T(z))=G(z)+1
\iL{abeleq}
\ee
Equations \rf{transfereq} and  \rf{abeleq} can be deduced from each other other by the change of variable.
Just replace $z$ to $F(z)$ in equation \rf{abeleq} and apply function $F$ to both side of the resulting equation.
The Readers are united to make this exercise or to look for it at some Wikipedia \cite{wikiabel}. I hope, the Soviet veterans, that remove articles about the USSR, will be stopped before they begin to vandalise the Abel equation.
\vskip 6pt

Assume, that the superfunction $F$ and the abelfunction $G$ are somehow constructed.
This gives key to the iterate $T^z$ of the transfer function, and the number $z$ of the  iteratie has no need to be integer.
This iterate can be expressed as follows:
\be
T^z(x)=F(z+G(x))
\eL{Tzx}
In order to show an example of application of formula \rf{Tzx}, return to the story about Manufacturer and Physicist from the previous section.
Assume, the Physicist has found the physically-meaningful solution $F$ of the transfer equation  \rf{transfereq}, and has constructed the inverse function, id est,
abelfunction $G\!=\!F^{-1}$.
Then, Physicist can express the transfer function $F$ of the amplifier of arbitrary length $z$ by formula \rf{Tzx}.
\vskip 6pt

For transfer function $T$, its superfunction $F$ and the abelfunction $G$ appear as two sponges of a wrench for some screw, as a tool, that allows to rotate the screw for any rational angle.

\section{Multiplicity of solutions}
\baselineskip 18.8pt

Solution $F$ of equation \rf{transfereq} is not unique.
One can reduce multitude to solution, specifying its value at zero (or at any other point the Reader likes);
id est, choose some number $F_0$ and add the requirement that
\be
F(0)=F_0
\eL{f0}
In many cases, the constant $F_0$ does not affect the shape of iterates of the transfer function; transform
\be
\tilde F(z)=F(z\!+\!x_0)
~,~ ~
\tilde G(z)=G(z)\!-\!x_0 ~
\eL{zo}
for some $x_0$ and substitution $F \!\rightarrow \!\tilde F$, $G\!\rightarrow \!\tilde G$ into equation \rf{Tzx} gives the same iterate, as the initial $F$, $G$ do.
\vskip 2mm

However, even after addition of requirement \rf{f0}, the solution is not unique. If $F$ is solution, superfunction, then another solution (another superfunction)
can be constructed with
\be
\tilde f(z)=F\big(z+\theta(z)\big)
\eL{tif}
where  $\theta$ is periodic holomorphic function with period unity.
At  $\theta(0)\!=\!0$, even condition \rf{f0} is preserved.
\vskip 2mm
The different superfunctions $F$, with their abelfunctions $G\!=\!F^{-1}$,
give
different iterates of the transfer function $T$. For this reason, one may think, that the non-integer iterates of a function have no meaning.
In particular, in century 20, the colleagues had believed, that the half iteration of factorial, denoted with $\sqrt{\,!\,}\!=\! \mathrm{Factorial}^{1/2}$, has no meaning.
\vskip 6pt

\begin{wrapfigure}{r}{144pt}
\begin{picture}(196,100)
\put(12,0){\sx{.33}{\ing{znachok}}}
\put(18,0){\sx{.33}{\ing{znachok}}}
\end{picture}
\vskip -6pt
\caption{ $\sqrt{\,!\,}$ as emblema
\label{SqrtFactorialLogo}
}\vskip -12pt
\end{wrapfigure}
Function $\sqrt{\,!\,}$ is especially interesting, because since 1950, it is used as logo of the Physics Department of the Moscow State University
\cite{kandi}. That logo is shown in figure
\ref{SqrtFactorialLogo}, borrowed from the Russian article \cite{sado}. Only in 2009, the physical and mathematical meaning of this iterate and this logo had been reported
\cite{facit},
when the apparatus of superfunctions had been constructed.
The uniqueness of the superfunction of factorial (and, therefore, that of the half iterate of factorial) is provided by the additional requirement of holomorphism and behaviour at infinity:
transformation \rf{tif} reduces the range of holomorphism of superfunction and that of the reconstructed non-initeger iterates.
\vskip 6pt

The problem of evaluation of non-integer iterate in some physically-meaningful way arises not only at the phenomenological consideration of an idealised amplifier.
Similar problem appears at the analysis of stability of jets; in the simple (single-dimensional) approximation, the appearance and disappearance of instability by the 
Pomeau-Manneville scenario \cite{pomeau,gac}  
can be described with some specific quadratic transfer function. Similar equation arise at the analysis of stability of attractors
\cite{Eckmann81}.
At least for the single-dimensional models, one can construct abelfunction and superfunction 
\cite{logistic}, and, hence, the non-integer iterates of the transfer function.
\vskip 2mm

Observation of similarities in construction of superfunctions allows to formulate the problems about superfunctions and the goal of this Book.
This is suggested in the next section.

\section{Formulation of the problem}
\baselineskip 17.6pt

Suppose we have some transfer function $ T $, holomorphic in a wide range of values of the argument, and we are interested in its iterates.
\vskip 6pt 

What additional conditions should be imposed on superfunctions $F$ (or to the abelfunction $G\!=\!F^{-1}$) 
as solution of the transfer equation \rf {transfereq} (or that of the Abel equation \rf {abeleq}) 
to provide the unique solution? 
\vskip 6pt 


How to evaluate the superfunctions and the abefunction, that satisfying the conditions chosen? 
\vskip 6pt 

How to verify that 
namely this solution has the physical meaning? 
\vskip 6pt 

The goal of this book is to answer these  questions.
I try to represent the answers in the most simple and explicit form. 
In this book, I retell the previously published 
articles 
\cite {analuxp, sqrt2, vladir, logistic, factorial, e1e, susin, sutra, 2013or};
the idea is to release the Reader from the need to drill the original publications.
\vskip 6pt 

The next chapter suggests examples of superfunctions and corresponding abelfunctions; other chapters suggest general algorithms for calculation superfunctions and the application to physical problems. Description of the state of a physical system with a single parameter is already a strong approximation, and at least in this case, it is desirable to represent the solution in a simple 
and exact 
form.
\vskip 6pt 

\begin{wrapfigure}{r}{150pt}
\vskip -8pt
\sx{.24}{\begin{picture}(300,740)
\put(0,-40){\ing{martini}}
\end{picture}}
\vskip -2pt
\caption{\large 
Dry martini
\label{martini}}
\vskip -4pt
\end{wrapfigure}
Perhaps, I should discuss meanings of term ``simple''. Applying to a function, this term may get new meaning, indicating, that values of the function belong to some finite set \footnote{\url{https://en.wikipedia.org/wiki/Simple_function} In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line, similar to a step function. }. In the similar way, the adjective ``dry'', being applied to a vine or Martini, significantly change its meaning
(figure \ref{martini}). In this Book, term ``simple'' indicates, that the function is easy to understand, its definition is short and its implementation is fast; as for the set of values, the functions are supposed to be holomorphic and, therefore, should have values from the continuous set of complex numbers. 
\vskip 6pt 

On this point I finish the general speculations about superfunctions and turn to the specific examples.
In the following chapter, the examples of transfer function and its superfunction and abelfunction  are considered.


\chapter{~ Examples of superfunctions \label{ExamplesOfSuperfunctions}}
\baselineskip 18pt

Before to calculate the superfunctions and abelfunctions for transfer functions of general kind,
it worth to see the cases, when the superfinction and the abelfunction can be easy expressed in terms of already known special functions.
These examples are considered in this chapter. It can be considered as continuation of the Inroduction: the most of functions mentioned here are known since the school course of algebra.
\vskip 2mm

Superfunction $F$ for the transfer function $T$ is solution to the transfer equation \rf{transfereq}; I repeat it here:
\be
\nonumber
F(z+1)=T(F(z))
\ee
In the next section, several known solutions are presented in Table \ref{TableOfSuperfunctions}.
Then, in the following sections and chapters, these functions are considered with more details.

\section{Table of superfunctions}

\begin{table}%\title{Table of SuperFunctions}
\begin{center}
\caption{Examples of superfunctions $f$: $~ ~T(z)=f\Big(1+g(z)\Big)$  \label{TableOfSuperfunctions}}
\newcommand \vv {\rule{0pt}{22pt}}
\hskip -12pt
\sx{.95}{\baselineskip 28pt\begin{tabular}{r|l|l|l|c}
&$\!T(z)\!
		$&$f(z)    		$&$g(z)=f^{-1}(z)            $& comment \\  \hline
1&$c				$&$ c			$&  & \vv \\			
2&$\!z\!+\!1		$&$b+z			$&$z-b			$&$b\in \bC		$ \vv\\
3&$b+z		$&$b z + c		$&$(z-c)/b				$&$b\ne 0		$ \vv\\
4&$b z + c		$&$b^z+\frac{c}{1-b}$&$\log_{b}\!\left(z-\frac{c}{1-b}\right) $& $b\!\ne\!0$, $b\ne 1$, \cite{log} \vv\\ 
5&$b^z 		$&${\rm tet}_b(z)		$&${\rm ate}_b(z)	$& \cite{analuxp,sqrt2,e1e} \vv\\
6&$z^b 		$&$\exp(b^z)		$&$\log_{b}\!\big(\ln(z)\big)	$&\!\!\rf{Pow}, $b\!>\!0$, $b\!\ne\! -1$ \vv\\ 
7&$-a^2/z		$&$\ds a \tan\Big(\frac{\pi}{2}z\Big)	$&$\ds \frac{2}{\pi}\arctan(z/a)	$&$ a>0$ \vv\\ 
8&$\ds \frac{z}{c+z}	$&$\ds \frac{1-c}{1-c^z}$&$\ds \log_c\Big(1-\frac{1-c}{z}\Big)	$&$ c\ne 0~$, $~ c\ne 1$ \vv\\ 
9$\vv$&$\ds \frac{z}{1+z}	$&$\!\!\ds 1/z$&$1/z$&$\!\!\!f\!=\!g$; $T^n(z)\!=\!\frac{z}{1+n z}\!$  \vv\\ 
$\!$10&$\ln(b\!+\!\rme^z)	$&$\ln(bz)		$&$\rme^z/b	$&$b\ne 0$  \vv\\
$\!$11&$(a^b\!+\!z^b)^{1/b}	$&$az^{1/b}		$&$(z/a)^b	$&$a\!>\!0$, $b\!\ne\!0	$ \vv\\
$\!$12&$2z \sqrt{1\!-\!z^2}	$&$\sin(\pi 2^z)		$&$\log_2\!\big(\arcsin(z)/\pi\big)\!\!$& \vv\\
$\!$13&$2z\sqrt{1\!+\!z^2}$&$\sinh(2^z)$&$\!\!\log_2\!\left(\ln\left(z+\sqrt{z^2\!+\!1}\!~\right)\!/\pi\right)\!\!\!$& \vv\\
$\!$14&$2z^2-1		$&$\cos(\pi 2^z)	$&$\log_2(\arccos(z))	$&  \vv\\
$\!$15&$2z^2-1		$&$\cosh(\pi 2^z)$&\!\!$\log_2\!\left(\ln\left(z+\sqrt{z^2\!-\!1}\!~\right)\!/\pi\right)\!\!$& \vv\\
$\!$16&$2z/(1\!-\!z^2)	$&$\tan(2^z)		$&$\log_2(\arctan(z))	$&  \vv\\
$\!$17&$2z/(1\!+\!z^2)$&$\tanh(2^z)$& $\ds\log_2\!\left(2\ln\!\left(\frac{z\!+\!1}{z\!-\!1}\right)\right)\!\!$&  \vv\\
$\!$18&$\fac(z)\!$&${\rm SuFac}(z)\!$&${\!\rm AuFac}(z)$&$\!\!\!$\rf{SuFac}, \rf{AuFac}; \cite{factorial}$\!$ \vv\\
$\!$19&$b\,z\,(1\!-\!z)$&$\!\! \mathrm{LogisticSequence}_b(z) \!\! $&$\!\mathrm{LogisticSequence}_b^{-1}(z)\!$&$\!\!$\rf{logilim}, \rf{AbelLogistic}; \cite{logistic}\! \vv\\
$\!$20&$\mathrm{Doya}^t(z) $&$ \mathrm{Tania}(tz) $&$(z+\ln(z)-1)/t $&$\!$\rf{Doya}, \rf{taniaprim} \vv\\% ; \cite{2013or,doya} \\
$\!$21&$\mathrm{Keller}^t(z) $&$ \mathrm{Shoka}(tz) $&$\mathrm{ArcShoka}(z)/t $&$\!$\rf{KellerDef}, \rf{Shoka} \vv\\% ; \cite{2013or,Keller,Shoka} \\
$\!$22&$\sin(z) $&$ \mathrm{SuSin}(tz) $&$\mathrm{AuSin}(z)/t $& \rf{susindef}, \cite{susin}  \vv\\
$\!$23&$\!\!\mathrm{zex}(z)\!=\!z \exp(z)\!\!$&$ \mathrm{SuZex}(tz) $&$\mathrm{AuZex}(z)/t $& \rf{zexDef} \cite{sutra}  \vv\\
$\!$24&$\!\!\mathrm{tra}(z)\!=\!z\!+\!\mathrm e^z\!\!$&$ \mathrm{SuTra}(tz) $&$\mathrm{AuTra}(z)/t $& \rf{tra} \cite{sutra}  \vv\\
\hline
&$P(T(Q(z)))\vv \!$& $P(f(z))$ & $g(Q(z))$ & $P(Q(z))\!=\!z$%, $P(Q(z))\!=\!z$
\\  \hline
\end{tabular}}
\end{center}
\end{table}

There is analogy between the table of superfunctions and tables of integrals. In both cases, the direct operation is, in certain sense, more difficult, than the inverse operation.
If one knows the indefinite integral of some function, and this integral is expressed in terms of elementary functions in a compact form (the writing fits the width of the column of the table), then, one can calculate the integrand with known rules of differentiation. In the similar way, if the superfunction $F$ and the abelfunction $G\!=\!F^{-1}$ are known, the 
transfer function can be expressed with
\be
T(z)=F(1+G(z))
\nonumber
\ee
One of ways to built-in the table of integrals is the building of table of derivatives.
One takes any short combination of basic elementary functions and differentiates it. If the result can be simplified to fit the width of the column of the table, then it is declared as ``function that can be analytically integrated''.
\vskip 2mm

In the similar way, one can deal with superfunctions.
Any elementary function $f$, 
for which the inverse function $g\!=\!f^{-1}$ 
also can be expressed as elementary function, 
can be taken as an example. Then, expression 
\be
t(z)=f(1+g(z))
\eL{tfg}
should be considered. Sometimes, after simplification, this expression fits the width of the column of the table. Then, this $t$ can be declared as 
``transfer function, for which the superfunction can be expressed analytically'', id est, also in terms of special functions. So, this 
$f$ is declared as its superfunction, and $g$ is declared as corresponding abelfunction. 
The most of table \ref{TableOfSuperfunctions} is built-up in such  a way. 
\vskip 2mm

Not all transfer functions can be expressed as elementary functions, and not all superfunctions can be expressed through special functions.
And not all abelfunctions. In this Book, I describe methods, how to deal with these cases. 
In general, any holomorphic function can be treated as transfer function, and the only edges of the range of holomorphism may limit the construction of the corresponding superfunction and the abelfunction. If the superfunction and the abelfunction are constructed, supplied with the specific names, described and implemented, they can be treated as a special functions. I call any function as ``special function'', if (and only if) the properties are revealed, described and the algorithm of the precise evaluation is supplied.
\vskip 2mm

Aiming the application in physics (and, perhaps, in other sciences), I am interested, first, in those functions, that can be evaluated %with required precision 
quickly. These functions can be used to construct new functions, and used to describe various phenomena. One can use them in the similar way, as one use other special functions (sin, bessel, erfc, etc.)
These functions appear in the Table \ref{TableOfSuperfunctions} with names; I supply also the number of formula or the cite,
to indicate, where the function is described, where can one find the algorithm for the evaluation. 
\vskip 2mm

Table \ref{TableOfSuperfunctions} appears as the basic toolbox for evaluation of non-integer iteration. 
In the following chapters, I describe, how the non-trivial superfunctions from the table can be constructed and evaluated.
However, 
first, it worth to check properties of elementary superfunctions. 

\section{Construction of elementary superfunctions}
\baselineskip 17pt

As it is mentioned above, searching for elementary superfunctions, it worth to begin not with a transfer function, but with the superfunction and the abelfunction,
applying formula \rf{tfg}. As an example, I show, how the 12th line of the Table \ref{TableOfSuperfunctions} can be verified in language Mathematica:
\vskip -8mm

~{\bf \normalsize
\begin{verbatim}
f[z_] = Sin[Pi 2^z]
g[z_] = Log[ArcSin[z]/Pi]/Log[2]
f[g[z]]
T[z_]=2 z Sqrt[1 - z^2]
Simplify[T[z] - f[1 + g[z]]]
\end{verbatim}
}
In Mathematica, the argument of function 
should appear
in squared brackets. No other trick, specific for Mathematica, is used; 
the verification
 can be performed in other languages too.
\vskip 6pt

\vskip 6pt

Table \ref{TableOfSuperfunctions} collects only the simplest (already described) superfunctions,
They can be modified, using the last row of the Table. Any pair of mutually inverse functions $P$ and $Q$
determines the transform, that can be applied to any of previous rows, giving the new transfer function with corresponding 
superfunction and abelfunction.
\vskip 6pt

The scaling transform relates the quadratic transfer function at rows 14 and 15 of the table with 
logistic operator (also quadratic transfer function) in raw 19, while parameter $b\!=\!4$; 
in this case, the logistic sequence is expressed with elementary function and its generalisation to the non-integer values of the argument is trivial \cite{logistic}. However, the superfunciton can be constructed also for other values of $b$; with these superfunctions,
the iterates dan be calculated. This case is considered below in chapter 7.
\vskip 2mm

A special case of transformation of superfunction is displacement of its argument for a constant.
In some cases, it is difficult to recognise this shift. For example, superfunctions in 14th and 15th rows of the table correspond to the same transfer function $ T(z)\!=\!2z^2\!-\!1$; these superfunction are related with translation of the argument for constant 
$\rmi \pi \ln(2)/2$. 
Similar relations take place also for other superfunctions. For superfunctions of $\exp_b$ for $1\!<\!b\!<\!\exp(1/\rme)~$ discussed in \cite{sqrt2}, this case is mentioned in Chapter 9.
\vskip 6pt

I invite the Reader to add any new rows to the table, following the trick above: choose some special function $f$, for which  $g\!=\!f^{-1}$ is also implemented, and try to simplify $t$, determined by equation \rf{tfg}.
\vskip 2mm

This chapter only declares the superfunctions.
Some elementary superfunctions are considered with more details in the next chapter.

\chapter{~ Elementary superfunctions}
\baselineskip 20pt

\begin{wrapfigure}{r}{136px}
\sx{.38}{
\begin{picture}(523,580)
\put(2,-8){\ing{Le-bourgeois-gentilhomme.jpg}}
\end{picture}}
\caption{\baselineskip 13pt
M. Jourdain: {\em These forty years now, I've been speaking in prose without knowing it!}  \cite{moliere}
\label{moliere}}
\end{wrapfigure}

In order to use superfunctions, described in this chapter, the Reader has no need to know, that they are superfunctions.
In the similar way, during 40 years, Joudrain has no need to know, that he speaks in prose
\cite{moliere}. However, many properties of elementary superfunctions are the same, as properties of other, nontrivial superfunctions, that cannot be easily expressed through elementary functions. In order to show these properties, in this chapter I consider elementary superfunctions.
\vskip 2mm

I would not like the colleagues to say, that {\em the formalism of superfunctions is too complicated}
\cite{akira2}. So I begin with very simple example, with linear function. 

\section{Iteration of linear function}

\begin{figure}[t] 
\hskip 12pt
\sx{.4}{\begin{picture}(960,970)
\put(0,0){\ing{itelin125}}
\put(480,984){\sx{3}{$y$}}
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\put(454,291){\sx{3}{$-2$}}
\put(454,191){\sx{3}{$-3$}}
\put(454,91){\sx{3}{$-4$}}
\put(70,475){\sx{3}{$-4$}}
\put(170,475){\sx{3}{$-3$}}
\put(270,475){\sx{3}{$-2$}}
\put(370,475){\sx{3}{$-1$}}
\put(495,475){\sx{3}{$0$}}
\put(595,475){\sx{3}{$1$}}
\put(695,475){\sx{3}{$2$}}
\put(795,475){\sx{3}{$3$}}
\put(895,475){\sx{3}{$4$}}
\put(984,476){\sx{3.1}{$x$}}
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\put(440,870){\rot{84}\sx{3.1}{$n\!=\!4$}\ero}
\put(470,870){\rot{82}\sx{3.1}{$n\!=\!3$}\ero}
\put(530,870){\rot{74}\sx{3.1}{$n\!=\!2$}\ero}
\put(616,870){\rot{65}\sx{3.1}{$n\!=\!1.2$}\ero}
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\put(724,870){\rot{57}\sx{3.1}{$n\!=\!0.6$}\ero}
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\put(889,712){\rot{32}\sx{3.1}{$n\!=\!-0.6$}\ero}
\put(889,671){\rot{29}\sx{3.1}{$n\!=\!-0.8$}\ero}
\put(889,633){\rot{26}\sx{3.1}{$n\!=\!-1$}\ero}
\put(905,518){\rot{13}\sx{3.1}{$n\!=\!-2$}\ero}
\put(900,424){\rot{2}\sx{3.1}{$n\!=\!-4$}\ero}
\put(874,392){\rot{.01}\sx{3.1}{$n\!\rightarrow\!-\infty$}\ero}
\end{picture}}
\vskip -2pt
\hskip 18pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Itelin125T.jpg} 
\vskip -4mm
\caption{\large
Iterates of linear function \rf{TABz} at $A\!=\!1$, $B\!=\!2~$;  $~y\!=\!T^n(x)$  versus $x$ for various $n$
\iL{itelin125}}
\vskip 8pt
\end{figure}
Consider the linear transfer function
\be T(z)=A+B z \eL{TABz}
where $A$ and $B$ are constants.
For $A\!=\!1$, $B\!=\!2~$, iterates of this function are shown in figure 
\ref{itelin125}.
Graphic $~y\!=\!T^n(x)$ is plotted versus $x$ for various values of the number $n$ of the iterate.
\vskip 2mm

The $n$th iterate of function $T$ can be written as follows:
\be
T^n(z)=A\frac{B^n-1}{B-1}+B^n z
\eL{TABzn}
This representation is used to plot figure \ref{itelin125}. 
The graphics %in this figure 
are straight lines. They cross at the point $(L,L)$ of the coordinate plane.
The fixed point $L$ is determined by the equation  $A+BL=L$, that gives
\be
L=A/(1-B)
\eL{Llin}
At $B\!\rightarrow \!1$,
the fixed point runs to infinity, and the graphics in the analogy of figure \ref{itelin125} become parallel.
For $A\!=\!1$, $B\!=\!2~$, this is case of figure  \ref{itelin125},
we get value 
$L\!=\!-1$; so, lines in figure \ref{itelin125} cross in point $(-1,-1)$.
\vskip 2mm

Representation \rf{TABzn} 
can be obtained with the general formula 
\rf{Tzx} 
with superfunction
\be
F(z)=A\frac{1-B^z}{1-B}
\eL{FTABz}
and abelfunction
\be G(z)=\log_b\Big(1+\frac{B-1}{A} z\Big)
\eL{GTABz}

In the special case $B\!=\!1$, 
the transfer function has no fixed points and representations \rf{TABzn}, \rf{FTABz}, \rf{GTABz} cannot be used.
For this case, id est, for $T(z)\!=\!A\!+\!z$, superfunctions and abelfunctions can be written as follows:
\be
F(z)=Az ~,~  ~  G(z)=z/A
\ee
Their combination gives
\be
T^n(z)=F(1+G(z))=A~(n+z/A)=An+z
\iL{Anz}
\ee
Those, who still think, that the formalism is too complicated 
\cite{akira2}, are invited to check the deduction above and verify, that this case corresponds to the level of a junior high school.
\vskip 2mm

\begin{wrapfigure}{r}{180px}
\sx{.25}{\begin{picture}(400,480)
\put(700,530){ \reflectbox{ \rot{-90}\ing{crack100}\ero} }
\end{picture}}
\caption{\large
%\normalsize
Mathematician uses \\[-8pt]
sledgehammer to crack a nut \label{izpishki}}
\end{wrapfigure}

To iterate the linear function  \rf{TABz}, 
the use of superfunction \rf{FTABz} 
and abelfunction \rf{GTABz} 
can be qualified with term ``use sledgehammer to crack a nut'', as it is shown in figure \ref{izpishki}.
However, this example is important: it is simple and it shows, how the superfunction and abelfunction 
can be used together to express iterates with formula \rf{Anz}. 

\section{Rational function}
\baselineskip 18pt

\begin{figure}[h]
\hskip 22pt
\sx{.2}{\begin{picture}(2002,2002) 
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\put(1588,-66){\sx{7}{$6$}} 
\put(1788,-66){\sx{7}{$8$}} 
\put(1958,-66){\sx{8}{$x$}} 
\put(0,0){\ing{f1xmap}} 
\put(0930,1666){\sx{8}{$u\!=\!0$}}
\put(1430,1666){\sx{8}{$v\!=\!0.1$}}
\put(1030,1490){\sx{8}{$v\!=\!0.2$}}
\put(1660,1430){\sx{8}{$u\!=\!-0.1$}} 
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\put(280,1120){\sx{8}{$v\!=\!0.2$}} 
\put(1480,1120){\sx{8}{$u\!=\!-0.2$}} 
\put(200,982){\sx{8}{$v\!=\!0$}} 
\put(1610,982){\sx{8}{$v\!=\!0$}} 
\put(640,438){\sx{8}{$v\!=\!-0.2$}}
\put(930,306){\sx{8}{$u\!=\!0$}}
\put(1410,176){\sx{8}{$v\!=\!-0.1$}} 
\end{picture}} 
\vskip 16pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:F1xmapT.jpg}
\vskip -4pt
\caption{\large $u\!+\!\mathrm i v\!=\! T(x\!+\!\mathrm i y)=1/(x\!+\!\mathrm i y)$ 
\iL{f1xmap}  }
\vskip 4pt
\end{figure}

\begin{figure}[h]
\sx{.42}{\begin{picture}(1004,1004)
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\put(882, 30){\rot{44}\sx{3}{$n\!=\!-1.7$}\ero}
\end{picture}}
\vskip -1pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Frac1zt.jpg}
\vskip -5pt
\caption{\large $y\!=\!T^n(x)$  for $T(z)\!=\!-1/z$ by formula \rf{linfrac1ite} at various $n$  \label{f1ziteplot}  }
\vskip 2pt
\end{figure}

The linear fraction, or ``rational function'' can be considered as generalisation of the linear function.
Let
\be
T(z)=\frac{U+V z}{W+z}
\eL{Tuvwz}
where  $U$, $V$ and $W$ are constant parameters. First, as an example, 
consider function
\be
T(z)=-1/z 
\eL{Tzm1z}
This example corresponds to $U\!=\!-1$, $V\!=\!0$, $W\!=\!0$ in formula \rf{Tuvwz}. Negative value of $U$ is chosen in order to have positive derivative at the positive values of the argument; iterates of a growing function are easy to interpret in terms of modelling of physical process. Iterates of decreasing function, contrary, imply dealing with complex values. I am interested mainly in the applications
for physics; so, I consider mainly the growing transfer functions
and growing real-holomorphic superfunctions.
\vskip 2mm

Compex map of function
$T$ by equation \rf{Tzm1z}
is shown in figure \ref{f1xmap}.
For this function, the levels of constant real part and levels of constant imaginary part are circles,
and all these circles pass through the origin of coordinates.
\vskip 2mm

For real values of the argument, iterates of function $T$ by \rf{Tzm1z} are shown in figure
\ref{f1ziteplot}. Lines $y\!=\!T^n(x)$ are plotted versus $x$ for various values of $n$. Below I describe, how these iterates are calculated.
\vskip 2mm

\be F(z)&=& \tan\Big(\frac{\pi}{2} z\Big) \\
G(z)&=&F^{-1}(z)= \frac{2}{\pi} \arctan\left(z\right) 
\ee
Then, the $n$th iterate of the transfer function $T^n(z)\!=\!F(n\!+\!G(z))$ appears to be
\be
T^{n}(z)= \frac{\ds -1-  \cot\left(\frac{\pi}{2} n \right)\, z}{\ds - \cot\left(\frac{\pi}{2} n\right)\, +\, z}
\eL{linfrac1ite}
This case is represented in row 7 of table
\ref{TableOfSuperfunctions}.
Expression \rf{linfrac1ite} is used to plot figure 
\ref{f1ziteplot}.
\vskip 2mm

\begin{figure}[t]
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\put(308, 0){\rot{10}\sx{2.9}{$n\!=\!-1$}\ero}
\end{picture}}
\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Fracit05t150.jpg}
\vskip -2pt
\caption{ $y=t^n(x)$ by \rf{tzcz} at  $c=0.5$ 
\iL{c05} }
\vskip 18pt
\end{figure}

Consider one more special case of formula \rf{Tuvwz}. Let
\be t(z)=\frac{z}{c+z} \eL{tzcz}
where $c$ is constant. Then
\be t^n(z)= \frac{z}{\ds c^n+\frac{1\!-\!c^n}{1\!-\!c} z}
\eL{tnzc}
Iterates of function $t$ for $c\!=\!0.5$ are shown in figure 
\ref{c05}. 
The same for  $c\!=\!1$ are shown in figure  \ref{c10},
and the same for  $c\!=\!2$ are shown in figure  \ref{c20}. 
Below I show, how these iterates are evaluated.
\vskip 2mm
\baselineskip 21pt
\begin{figure}[t]
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\put(686,75){\sx{3}{$x$}}
\put(0,0){\ing{fracit10}}
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\put(606,260){\rot{6}\sx{3}{$n\!=\!0.4$}\ero}
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\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Fracit10t150.jpg}
\caption{ $y\!=\!t^n(x)$ by \rf{tzcz} at  $c\!=\!1$ 
\iL{c10} }
\vskip 18pt
\end{figure}

\begin{figure}[t]
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\end{picture}}
\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Fracit20t150.jpg}
\vskip -6pt
\caption{\large $y\!=\!t^n(x)$ by \rf{tzcz} at  $c\!=\!2$  \iL{c20} }
\vskip 18pt
\end{figure}
\vskip 2mm
For the transfer function $t$ by \rf{tzcz}, superfunction $f$ can be written as follows: %в таком виде:
\be
f(z)=\frac{c-1}{c^z-c}
\eL{fzfrac}
Corresponding Abel function $g=f^{-1}$ 
is
\be
g(z)=\log_c\Big(1+\frac{c\!-\!1}{z}\Big)
\eL{gzfrac}
Expressions \rf{fzfrac} and \rf{gzfrac} are used to evaluate $t^n(z)\!=\!f(n\!+\!G(z))$ and plot figures, in particular,
figures 
\ref{c05},
\ref{c10} and
\ref{c20}. The Reader is invited to plot iterates of function $t$ by formula \rf{fzfrac} at other values of $c$ too.
\vskip 2mm
The iterates of the rational function of real argument show the smooth transition from function $t$ to the identity function and then to inverse function $t^{-1}$.
All the curves in figures 
\ref{c05},
\ref{c10} and
\ref{c20} pass through the fixed point. This value is mapped to itself at the iterates of the transfer function. This property is not specific feature of the rational function. Other functions, considered on the following chapters, have the same property.

\vskip 2mm
For construction and uniqueness of superfunctions in the following sections, it is important to consider them in the complex plane. The complex maps help to understand properties of these functions. Following the idea to begin with simple examples, I suggest the complex maps of superfunction $f$ and abelfunction $g$.
For $c\!=\!2$, the complex map of superfunction $f$ by formula \rf{fzfrac} is shown in figure 
\ref{fzfracmap}. Similar map of the Abel function $g$ is shown in figure
\ref{gzfracmap}. 

\begin{figure}[t]
\hskip 22pt
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\put(0,0){\ing{fz2z1map}} 
\put(310,1888){\sx{8}{$v\!=\!0$}}	 \put(1710,1888){\sx{8}{$v\!=\!0$}} 
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\put(310,1432){\sx{8}{$v\!=\!0$}} 
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\put(310,984){\sx{8}{$v\!=\!0$}}	 \put(1710,980){\sx{8}{$v\!=\!0$}} 
\put(1710,758){\sx{8}{\rot{-1}$u\!=\!0$\ero}} %
\put(310,522){\sx{8}{$v\!=\!0$}}	
\put(310,748){\sx{8}{$u\!=\!-0.5$}}	 \put(1710,300){\sx{8}{$u\!=\!0$}}%
\put(310,300){\sx{8}{$u\!=\!-0.5$}} 
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\put(1300,1360){\sx{8}{$u\!=\!-0.1$}}
\put(1340,1120){\sx{8}{$v\!=\!-0.1$}}
\put(1360,1020){\sx{8}{$u\!=\!0.1$}}
\put(1334,850){\sx{8}{$v\!=\!0.1$}}
\end{picture}} 
\vskip 12pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Fz2z1mapT.jpg}
\vskip -6pt
\caption{\large
$u\!+\!\mathrm i v=f(x\!+\!\mathrm i y)$
by \rf{fzfrac} at $c\!=\!2$
\iL{fzfracmap}}
\end{figure}

\begin{figure}[h]
\hskip 22pt
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\put(1788,-66){\sx{7}{$8$}} 
\put(1958,-66){\sx{8}{$x$}} 
\put(0,0){\ing{gz2z1map}} 
\put(880,1920){\sx{8}{$u\!=\!0$}} 
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\put(1040,250){\sx{8}{$v\!=\!0.2$}}
\put(250,232){\sx{8}{$u\!=\!-0.1$}} \put(1740,260){\sx{8}{$u\!=\!0.1$}} 
\put(210,110){\sx{8}{$v\!=\!0.1$}} \put(1610,110){\sx{8}{$v\!=\!0.1$}} 
\put(880,120){\sx{8}{$u\!=\!0$}} 
\end{picture}} 
\vskip 12pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Gz2z1mapT.jpg}
\vskip -6pt
\caption{\large
$u\!+\!\mathrm i v=g(x\!+\!\mathrm i y)$%right, plotted by \rf{fzfrac} and \rf{gzfrac};
by \rf{gzfrac} at $c\!=\!2$
\iL{gzfracmap}}
\end{figure}

\vskip 2mm

Function $f$ by formula \rf{fzfrac} is periodic; its period
\be
P=2 \pi \mathrm i / \ln(c)
\ee
At $c\!=\!2$, this period $P \!=\! 2 \pi \mathrm i/ \ln(2) \!\approx\! 9.06472 \,\mathrm i$.
Vertical size of figure \ref{fzfracmap} covers a little bit more that two periods of this function.
For real $c$, the period is pure imaginary; the map reproduces itself at the translations for integer factor of $|P|$ along the imaginary axis.
\vskip 2mm

\baselineskip 18.4pt
The inverse function $g\!=\!f^{-1}$ is expressed by formula \rf{gzfrac}.
It is Abel function for the transfer function $t$.
Abelfunction $g$ is shown with its complex map in figure  \ref{gzfracmap} 
for the same value of parameter, $c\!=\!2$, as function $f$ in figure  \ref{fzfracmap} 
\vskip 2mm

Function $g$ has two branch points,  $c\!-\!1$ and zero.
For real $c$, the cut line between these two points belongs to the real axis.
Equilnes are symmetric with respect to reflections from line 
$\Re(x)=(c\!-\!1)/2$.
\vskip 2mm

With superfunction  $f$ and abelfunction $g$ by formulas 
\rf{fzfrac} and \rf{gzfrac}, iterates of the transfer function $t$ can be written as usually,
\be
t^n(z)=f(n+g(z))
\eL{tnzfg}
The readers are invited to check, that this representation agree with expression \rf{tnzc}.
It is better to do this exercise for the simple. % case of linear fraction.
Then it will be easier to reproduce the similar exercises for other superfunctions in the following sections.
\vskip 2mm

\section{Power function and its iterates}
\baselineskip 19pt

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\put(1026,-9){\sx{3.3}{$x$}}
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\end{picture}}
\vskip 2pt
\hskip 25pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Z2itmapT.jpg}
\vskip -6pt

\caption{\large $u\!+\!\mathrm i v=T(x\!+\!\mathrm i y) =(x\!+\!\mathrm i y)^2$,
by \rf{Pow} for $a\!=\!2$
\iL{z2itmap}
}
\vskip 18pt
\end{figure}
\begin{wrapfigure}{r}{110px}
\sx{.24}{\begin{picture}(480,340)
\put(20,-30){\ing{sam107}}
\end{picture}}
\caption{Superpower$\!\!\!\!\!\!\!\!\!\!$ \label{osam}}
\end{wrapfigure}
Consider the power function
\be \!\!\!\!
T(z)=\mathrm{Pow}_a(z)=z^a ~ ~
\eL{Pow}
For $a\!=\!2$, the map of function $T$ 
is shown in figure \ref{z2itmap}.
Here, I suggest the superfunction of the power function, id est, the superpower function.
(However, term ``superpower'' may have also other meaning, as it is shown in figure \ref{osam}, but such a meaning is not used in this Book).
\vskip 8pt

Consider iterates of the power function $T$ by \rf{Pow}, they can be written as follows:
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!
T^n(z) =
{\mathrm{Pow}_a}^n(z)=z^{a^n}= 
\underbrace
{\mathrm{Pow}_a\Big( \mathrm{Pow}_a \big(... \mathrm{Pow}_a(z).. \big)\Big)}
_{\ds n \text{ evaluations of function Pow}_a}
\label{Pown} 
\ee

\vskip 2mm

Figure \ref{IterPowPlot} shows  $T^n$ by formula \rf{Pown} for $a\!=\!2$ at various  $n$.
\begin{figure}[t]
\sx{.4}{\begin{picture}(1002,1012) 
\put(10,10){\ing{IterPowPlot}}
\put(11,976){\sx{4}{$x^{2^n}$}} 
\put(11,898){\sx{4}{$9$}} 
\put(11,798){\sx{4}{$8$}} 
\put(11,698){\sx{4}{$7$}} 
\put(11,598){\sx{4}{$6$}} 
\put(11,498){\sx{4}{$5$}} 
\put(11,398){\sx{4}{$4$}} 
\put(11,298){\sx{4}{$3$}} 
\put(11,198){\sx{4}{$2$}} 
\put(11,098){\sx{4}{$1$}} 
\put(100,16){\sx{4}{$1$}} 
\put(200,16){\sx{4}{$2$}} 
\put(301,16){\sx{4}{$3$}} 
\put(401,16){\sx{4}{$4$}} 
\put(502,16){\sx{4}{$5$}} 
\put(602,16){\sx{4}{$6$}} 
\put(703,16){\sx{4}{$7$}} 
\put(803,16){\sx{4}{$8$}} 
\put(903,16){\sx{4}{$9$}} 
\put(990,16){\sx{4}{$x$}}  
\put(112,850){\sx{3.8}{\rot{90}$n\!\rightarrow \infty $\ero}} 
\put(152,866){\sx{3.8}{\rot{88}$n\!=\!3$\ero}} 
\put(191,870){\sx{3.8}{\rot{87}$n\!=\!2$\ero}} 
\put(325,921){\sx{3.8}{\rot{81}$n\!=\!1$\ero}} 
\put(461,893){\sx{3.6}{\rot{70}$n\!=\!0.6$\ero}} 
\put(509,894){\sx{3.6}{\rot{69}$n\!=\!0.5$\ero}} 
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\put(826,913){\sx{3.6}{\rot{50}$n\!=\!0.1$\ero}} 
\put(928,932){\sx{3.8}{\rot{45}$n\!=\!0$\ero}} 
\put(894,778){\sx{3.6}{\rot{39}$n\!=\!-0.1$\ero}} 
\put(888,676){\sx{3.6}{\rot{33}$n\!=\!-0.2$\ero}} 
\put(884,596){\sx{3.6}{\rot{28}$n\!=\!-0.3$\ero}} 
\put(877,528){\sx{3.6}{\rot{24}$n\!=\!-0.4$\ero}} 
\put(871,476){\sx{3.6}{\rot{19}$n\!=\!-0.5$\ero}} 
\put(870,429){\sx{3.6}{\rot{17}$n\!=\!-0.6$\ero}} 
\put(890,297){\sx{3.8}{\rot{8}$n\!=\!-1$\ero}} 
\put(884,172){\sx{3.8}{\rot{1}$n\!=\!-2$\ero}} 
\put(881,132){\sx{3.8}{\rot{.3}$n\!=\!-3$\ero}} 
\put(852,100){\sx{3.6}{\rot{.1}$n\!\rightarrow - \infty$\ero}} 
\end{picture}} % 
\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:IterPowPlotT.jpg}
\vskip -4pt
\caption{\large
$y\!=\!T^n(x)\!=\!\mathrm{Pow_2}^n(x)$ для различных $n$. 
\iL{IterPowPlot}
\label{IterPowPlotA}}
\vskip 4mm
\end{figure}
Iterates of the power function can be expressed also with the general formula,
\be %\nonumber
T^n(x)=F\Big(n+G(x)\Big)
\eL{againTc}
where $F$ is superfunction of the transfer function $T$ and $G\!=\!F^{-1}$ is the abelfunction.

For the power function, the superfunction (id est, the superpower function)
can be written as follows:
\be F(z)=\exp\Big(\exp(\ln(a)\, z)\Big) = \exp^2\big(\ln(a)\,z\big) \eL{powF}
Inverting this representation, one can get the abelfunction; id est, the ``abelpower'' function,
\be 
G(z)=\ln\Big(\ln(z)\Big)/\ln(a) = \ln^2(z)/\ln(a) 
\eL{powG} 
These formulas correspond to the 6th row of table 
\ref{TableOfSuperfunctions}.
\vskip 2mm


For the transfer function $T$ by \rf{Pow}, iteration \rf{again} can be simplified,
\be
T^n(z) =\exp^2\Big(\ln(a)\big(n+\ln^2(z)/\ln(a)\big)\Big)=z^{a^n}
\eL{zbc}
leading to the 3d expression in equation \rf{Pown}.

In such a way, iterates of the transfer function $T$ by \rf{Pow} 
appear to be function of the similar kind (also power function).
I invite the Reader to plot the complex maps of superpower function $F$ by \rf{powF}, abelpower function $G$ by \rf{powG} and iterates of power function by \rf{zbc}.  
\vskip 2mm

\JP{
Формулы этой главы, как, впрочем, и последующих глав, выписываются подробно с тем, чтобы создать условия для конструктивного обсуждения, чтобы такое обсуждение не выглядело как разгром, показанный на рисунке \ref{grom}.

~

\begin{figure}[h]
\sx{.5}{\ing{Gromit}}
\caption{Конструктивные формулы должны допускать опровержение, с тем, чтобы их обсуждение носило академический характер \iL{grom}}
\end{figure}
}

For  $T(z)\!=\!z^a$, the simple relations take place:
\be
T(z^b)=T(z)^b
\eL{fnz1}
At $b\!=\!a\!=\!2$, in addition, the following relation hakes place:
 \be
T^b(z)=T(z)^b
\eL{fnz2}

There is common confusion (the wrong public opinion), that some simple equivalents of relations
\rf{fnz1}, \rf{fnz2} 
should take place for other values of $b$ and for other functions too. 
In particular, one of critics of the publication about half iteration of factorial \cite{facit} had insisted, that 
$\mathrm{Factorial}^{1/2}(z)=\mathrm{Factorial}(z)^{1/2}$, and I was not successful explaining, that it is not the case, even if notation
$\sqrt{\mathrm{Factorial}}=\mathrm{Factorial}^{1/2}$ is used. 
The factorial is considered in this Book in the Chapter 8. However, before to deal with factorial, I would like to consider simple example. The power function is one of the simple examples.
I invite the Reader to check, that even for $a\!=\!3$, relation \rf{fnz2} is not valid (except of some specific values of $z$).

\JP{

\section{Extensions}

In this chapter, few examples of superfunctions are suggested, namely, the linear function, the rational function and the power function.
In these cases, both, superfunctions and the abelfunctions can be expressed in terms of the elementary functions.
\vskip 2mm

Dealing with elementary superfunctions, one has no need to know, that they are superfunctions. 
Formulas for the examples from this chapter can be extended, using the transform at the bottom of
\ref{TableOfSuperfunctions}. However, the same transform, extension apply also to other superfunctions, not only to the elementary superfunctions.
}
\vskip 2mm
Having the explicit representation, one can calculate all the iterates necessary, including the non-integer iterates. But not so many elementary functions are inverse of some other elementary functions,
Perhaps, it is better to refer to special functions.
The simple example of the non-elementary function is considered in the next chapter.


\chapter{~Tania and Shoka}
\baselineskip 19.2pt

\begin{figure}[h]
\vskip -26pt
\begin{center}
\sx{.44}{%\ing{ShokoTania}}
\begin{picture}(802,400)
\put(0,0){\includegraphics{ShokoTania}}
\put(380,380){\sx{3.2}{$y$}}
\put(380,292){\sx{3.2}{$3$}}
\put(380,192){\sx{3.2}{$2$}}
\put(380, 92){\sx{3.2}{$1$}}
\put( 75,-34){\sx{3.2}{$-\!3$}}
\put(175,-34){\sx{3.2}{$-\!2$}}
\put(275,-34){\sx{3.2}{$-\!1$}}
\put(394,-34){\sx{3.2}{$0$}}
\put(494,-34){\sx{3.2}{$1$}}
\put(594,-34){\sx{3.2}{$2$}}
\put(694,-34){\sx{3.2}{$3$}}
\put(784,-34){\sx{3.2}{$x$}}
\put(442,140){\sx{3.2}{\rot{41} $y\!=\!\mathrm{Shoka}(x)$\ero}}
\put(468,102){\sx{3.2}{\rot{32} $y\!=\!\mathrm{Tania}(x)$\ero}}
\end{picture}}
\vskip 16pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ShokotaniaT.png}
\end{center}
\vskip -18pt
\caption{
Functions Tania and Shoka by \rf{shoka0} and \rf{Tania0}
\iL{shokataniaplot}
\label{ShokaTaniaplot}}
\vskip -4pt
\end{figure}

This chapter considers functions, that have applications in laser science. I call them Tania and Shoka. For some vicinity of the real axis, these functions can be defined as follows:
\be
\mathrm{Shoka}(z)&=&z+\ln(\mathrm e^{-z}\!+\!\mathrm e\!-\!1)  \iL{shoka0}\\
\mathrm{Tania}(z)&=&\mathrm{WrightOmega}(z\!+\!1)  \iL{Tania0}\label{TaniaWright}
\ee
At
$|\Im(z)|\!\ge\! \pi$, 
function
$\mathrm{WrightOmega}(z)$ behaves in a way I dislike. To avoid confusions, for the function I like, I use name Tania.
Below I define functions Shoka and Tania for the complex argument.
However, the representations \rf{shoka0} and \rf{Tania0}
are sufficient to plot Shoka and Tania in figure \ref{ShokaTaniaplot}.
\vskip 2mm



This chapter retells the contents of articles \cite{thailand,2013or}. 
This chapter should be especially important for the narrow specialists, who works in the laser science.
\JP{
\vskip 2mm\begin{figure}[h]
\begin{center}
\sx{.5}{\begin{picture}(400,240)
\put(0,-40){\ing{taniashoka}}
\end{picture}}
\end{center}
\caption{Таня и Шока \iL{taniashokadraw}}
\vskip -8pt
\end{figure}
}%JP

\JP{
\section{Об узких специалистах}
\baselineskip 19pt
Предупреждение. Узким специалистам лучше пропустить этy секцию: она содержит критическое мнение об узких специалистах и может оскорбить их религиозные чувства.
\vskip 2mm

\begin{wrapfigure}{r}{90pt}
\sx{.7}{\begin{picture}(80,290)
\put(8,10){\ing{Drink02}}
\end{picture}}
\caption{Узкий специалист \label{drink}}
\end{wrapfigure}

Узкие специалисты подобны инопланетянину зайцу Бо, которому кажется, что все только и думают, как бы сказать чего-нибудь нехорошее про зайцев
\footnote{ \url{https://www.youtube.com/watch?v=OCndqQqHlQo} Магазинчик Бо. Эпизод 23. Сень Бо-Гэ. Mar 1, 2013. {\em А ещё он говорил что-то нехорошее про зайцев!..}}. Известный классик Русской литературы Козьма Прутков обнаружил и указал, что узкий специалист подобен флюсу: его полнота односторонняя. Узкий специалист может легко понять и даже проделать вычисления, если он видит за ними реалистичную жизненную ситуацию. Например, человек может встретить трудности, складывая два числа, 250 и 250. Однако, если ему сказать, что речь идет о граммах водки, он мгновенно проведет необходимые рассчеты и сразу скажет ответ: {\em ``Да это же Поллитра!''}; последнее слово будет произнесено с уважением и придыханием. Это придыхание укажет присутствующим, 
что он любитель выбить, и вероятно, даже не просто любитель, а  профессионал и, возможно, даже узкий специалист (рис. \ref{drink}). Аналогично, узкому специалисту-лазерщику трудно понять, что такое ``передаточная функция'', до тех пор, пока ему не скажут, что некоторый параметр (мощность, интенсивность, флюенция или энергия светового импульса) на выходе лазерного усилителя рассматривается как функция аналогичного параметра на входе в усилитель.
\vskip 3mm

Функции Дойя и Келлер, определяемые в этой главе ниже, могут быть интерпретированы как передаточные функции
идеализированного
лазерного усилителя. Читатели, равнодушные к лазерной науке, могут рассматривать эти функции как абстрактные математические объекты; или даже вовсе пропустить эту главу, за исключением, может быть, секции  5 (``О единственности решения''): секция 5 может помочь тем, кто собирается раскритиковать эту Книгу или даже просто сказать что-нибудь нехорошее про лазерщиков.

}%JP

\section{Tania and Arctania}

\begin{figure}[t]
\begin{center}
\sx{2.5}{\begin{picture}(172,168)
\put(6,5){\ing{taniamap}}
\put(2,162){\sx{.7}{$y$}}
\put(2,144){\sx{.6}{$6$}}
\put(2,124){\sx{.6}{$4$}}
\put(2,104){\sx{.6}{$2$}}
\put(3,116){\sx{.6}{$\pi$ ~ \bf cut}}
\put(23,100){\sx{.8}{$u\!=\!0$}}
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\put(20, 84){\sx{.8}{$v\!=\!0$}}
\put(23,68){\sx{.8}{$u\!=\!0$}}
\put(-3,64){\sx{.6}{$-2$}}
\put(-3,53){\sx{.6}{$-\pi$ ~ \bf cut}}
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\put(-3,24){\sx{.6}{$-6$}}
\put( 22,0){\sx{.6}{$-6$}}
\put( 42,0){\sx{.6}{$-4$}}
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\put(134, 97){\rot{-4}\sx{.8}{$v\!=\!1$}\ero}
\put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}
\put(134, 72){\rot{3}\sx{.72}{$v\!=\!-\!1$}\ero}
\put(133, 60){\rot{3}\sx{.72}{$v\!=\!-\!2$}\ero}
\put(134, 48){\rot{3}\sx{.72}{$v\!=\!-\!3$}\ero}
\put(135, 36){\rot{3}\sx{.72}{$v\!=\!-\!4$}\ero}
\put(136, 25){\rot{3}\sx{.72}{$v\!=\!-\!5$}\ero}
\put(137, 14){\rot{3}\sx{.72}{$v\!=\!-\!6$}\ero}
\end{picture}}
\end{center}
\vskip -12pt
\url{https://mizugadro.mydns.jp/t/index.php/File:TaniaContourPlot100.png}
\vskip -4pt
\caption{ Complex map of function Tania, 
$~u\!+\!\mathrm i v\!=\!\mathrm{Tania}(x\!+\!\mathrm i y)$
\iL{TaniaMap}
\label{figTaniaMap}
}
\vskip 12pt
\end{figure}

Let function Tania be solution $f$ of the differential equation
\be \!\!\!\!\!
f'(z)= \frac{f(z)}{1+f(z)}
\eL{taniaprim}
with 
additional
condition
$f(0)\!=\!1$,
where contour of integration of
\rf{taniaprim} for $f(z)$
goes first from zero to imaginary part of $z$ along the imaginary axis, and then, along the line, parallel to the real axis, goes to point $z$.
Figure \ref{ShokaTaniaplot} shows function Tania of real argument. For moderate values of imaginary part of the argument, 
solution $f\!=\!\mathrm{Tania}$ of equation \rf{taniaprim} is expressed through the special function WrightOmega with equation \rf{Tania0}.  Complex map of function Tania is shown in figure \ref{TaniaMap}. 
\vskip 2mm

For real values of argument, Tania is positive and grows monotonously.
Toward the negative values of the argument, Tania decays exponentially.
At zero, Tania grows with tangent 1/2.
At large positive values of the argument, Tania grows in a way, similar to the linear function with tangent unity.

\vskip 2mm

Function Tania, shown in figure
\ref{figTaniaMap}, has two branch points, $-2\pm \mathrm i \pi$.
The cut lines are directed parallel to the real axis toward its negative direction;
they are determined by specification of path of integration of equation \rf{taniaprim}.
\vskip 2mm

\begin{figure}[t]
\begin{center}
\sx{2.5}{\begin{picture}(162,168)
\put(6,5){\ing{arctaniamap}}
\put(2,162){\sx{.7}{$y$}}
\put(2,144){\sx{.6}{$6$}}
\put(2,124){\sx{.6}{$4$}}
\put(2,104){\sx{.6}{$2$}}
\put(2, 84){\sx{.6}{$0$}}
\put(59,85){\sx{.6}{\bf cut}}
\put(-3,64){\sx{.6}{$-2$}}
\put(-3,44){\sx{.6}{$-4$}}
\put(-3,24){\sx{.6}{$-6$}}
\put( 22,0){\sx{.6}{$-6$}}
\put( 42,0){\sx{.6}{$-4$}}
\put( 62,0){\sx{.6}{$-2$}}
\put( 86,0){\sx{.6}{$0$}}
\put(106,0){\sx{.6}{$2$}}
\put(126,0){\sx{.6}{$4$}}
\put(146,0){\sx{.6}{$6$}}
\put(164,0){\sx{.7}{$x$}}
\put( 81.4, 23){\rot{81}\sx{.8}{$u\!=\!0$}\ero}
\put( 91.6, 23){\rot{81}\sx{.8}{$u\!=\!1$}\ero}
\put(101, 22){\rot{81}\sx{.8}{$u\!=\!2$}\ero}
\put(111, 21){\rot{83}\sx{.8}{$u\!=\!3$}\ero}
\put(120.4, 21){\rot{84}\sx{.8}{$u\!=\!4$}\ero}
\put(139,155){\rot{4}\sx{.8}{$v\!=\!8$}\ero}
\put(138,146){\rot{4}\sx{.8}{$v\!=\!7$}\ero}
\put(138,136){\rot{4}\sx{.8}{$v\!=\!6$}\ero}
\put(138,127){\rot{4}\sx{.8}{$v\!=\!5$}\ero}
\put(137,118){\rot{4}\sx{.8}{$v\!=\!4$}\ero}
\put(136,109){\rot{4}\sx{.8}{$v\!=\!3$}\ero}
\put(135,100){\rot{4}\sx{.8}{$v\!=\!2$}\ero}
\put(134, 92){\rot{3}\sx{.8}{$v\!=\!1$}\ero}
\put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}
\put(134, 76){\rot{-3}\sx{.8}{$v\!=\!-\!1$}\ero}
\put(133, 68){\rot{-5}\sx{.8}{$v\!=\!-\!2$}\ero}
\put(134, 59){\rot{-5}\sx{.8}{$v\!=\!-\!3$}\ero}
\put(135, 51){\rot{-5}\sx{.8}{$v\!=\!-\!4$}\ero}
\put(135, 41){\rot{-5}\sx{.8}{$v\!=\!-\!5$}\ero}
\put(135, 32){\rot{-5}\sx{.8}{$v\!=\!-\!6$}\ero}
\put(136, 23){\rot{-5}\sx{.8}{$v\!=\!-\!7$}\ero}
\put(137, 14){\rot{-5}\sx{.8}{$v\!=\!-\!8$}\ero}
\end{picture} } 
\url{https://mizugadro.mydns.jp/t/index.php/File:ArcTaniaMap.png}
\end{center}
\vskip -16pt
\caption{ \large
$u\!+\!\mathrm i v\!=\!\mathrm{ArcTania}(x\!+\!\mathrm i y)$
\label{figArcTaniaMap}
\iL{ArcTaniaMap}
}
\vskip 6pt
\end{figure}
\vskip 2mm

Complex map of the inverse function, id est, 
$\mathrm{ArcTania}\!=\!\mathrm{Tania}^{-1}$ is shown in figure \ref{ArcTaniaMap}. It can be expressed as elementary function,
\be
\mathrm{ArcTania}(z)=z+\ln(z)-1 ~ ~ ~
\eL{ArcTaniaz}
There are no complex constants in the representations of functions Tania and Arctania;
these functions are real-holomorphic:
\be 
\mathrm{Tania}(z^*)=\mathrm{Tania}(z)^* ~,~~ \mathrm{ArcTania}(z^*)=\mathrm{ArcTania}(z)^*
\label{Taniastar}
\ee

Functions Tania and ArcTania look similar to the linear function at large values of the argument; the lines of levels of constant real part and those of constant imaginary part form almost rectangular grid. Functions ArcTania 
also have
almost linear asymptotic.

\vskip 2mm

The almost linear asymptotic behaviour of 
$\mathrm{Tania}(z)$ at large %values 
$|z|\!\gg\! 1$
holds for the most of the complex plane, except the strip $\Re(z) \!<\!0$,  $|\Im(z)|\!\le\! \pi$.
In the strip   $|\Im(z)|\!<\! \pi$, at $\Re(z) \!\rightarrow\! -\infty$, function $\mathrm{Tania}(z)$ 
decays exponentially, this agrees with graphic of this function for real argument at figure \ref{shokataniaplot}.
\vskip 2mm

For moderate values of the imaginary part of the argument, function
$\mathrm{Tania}$ 
can be expressed through the known special function WrightOmega
\cite{wright,david} with equation \rf{Tania0}.
\JP{
Есть несколько оправданий для введения новой функции, которая легко выражается через известную функцию:

1. Для 
удобства оценок, желательно, чтобы суперфункция в нуле принимала целое значение,
Функция Tania будет такой суперфункцией для примера, рассматриваемого ниже. 

2. Имя Tania короче, чем WrightOmega, и допускает простую русскоязычную транслитерацию ``Таня''. При этом появляется возможность склонять имя этой функции по падежам в соответствии с традициями грамматики русского языка. 

3. В комплексной плоскости Таня как решение уравнения \rf{taniaprim} ведет себя проще, чем функция WrightOmega, представленая в алгоритмических языках программирования. 

Использование нового имени позволит избежать  путаницы, если одновременно потребуются обе функции, функция WrightOmega со счетным множеством разрезов и функция Таня с её двумя разрезами, видными на левой карте рисунка \rf{figTaniaMap}. Читатели приглашаются построить карты обеих функций, Тани и ВрайтОмеги\cite{wright,david}, и проверить, что только в полосе вдоль вещественной оси эти функции получаются одна из другой трансляцией аргумента на единицу.
\vskip 2mm
}
In particular, one has no need to make any difference between $\mathrm{Tania}(z)$ and $\mathrm{WrightOmega}(z\!+\!1)$
for real %values of 
$z$.
\vskip 2mm

For the efficient evaluation of Tania, its asymptotic expansions can be used. The whole complex plane can be covered with these expansions.
\vskip 2mm

At large values of the argument, Tania can be expanded as follows:
\be \!\!\!\!\!\!\!\!
\mathrm{Tania}(z)=z + 1\!-\!\ln(z) 
+\frac{ \ln(z)\!-\!1}{z} 
+\frac{\ln(z)^2\!-\!4\ln(z)\!+\!3}{2z^2} + ..
\label{TaniaBigz}
\ee
The effective small parameter of the expansion \rf{TaniaBigz} is 
$\ln(z)/z$.
For negative values of 
$\Re(z)$, 
this expansion is valid while
$|\Im(z)| \!>\! \pi$
However, this representation does not work between the cut lines at figure  \ref{TaniaMap}.
For the representation of Tania in this half-strip, I define the new variable
$\varepsilon=\exp(1\!+\!z)$; 
then Tania can be expanded as follows:
\be
\mathrm{Tania}(z) =
 \varepsilon
-\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{8}{3}\varepsilon^4+\frac{125}{24}\varepsilon^5+ O(\varepsilon^6)
\label{TaniaMezdu}
\ee
\vskip 2mm

The expansions above are not good %at all
for the moderate values of the argument, and especially poor is the approximation in vicinity of the branch points. For the branch point, the following expansion takes place: 
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{Tania}(z)=
-1+3t-3t^2
+\frac{3}{4} t^3
+\frac{3}{10}t^4
+\frac{9}{160}t^5
+..
\label{TaniaL}
\ee
where ~
$\ds
t= \mathrm{i} \sqrt{ \frac{2}{9} \big(z\!+\!2\!-\!\pi \mathrm i \big)~}
$.
\vskip 2mm

In addition, one may use the Taylor expansion at zero:
\be\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{Tania}(z)=1 
+\frac{z}{2}
+\frac{z^2}{16}
-\frac{z^3}{192}
-\frac{z^4}{3072}
+\frac{13 z^5}{61440}
-\frac{47 z^6}{1474560}
+ ..
\label{TaniaTay}
\ee
\vskip 2mm

With the representations above, for every point of the complex plane, for $\mathrm{Tania}(z)$,  one can get the ``zeroth'' approximation, let it be called $s_0$, with few correct significant figures.
Then, in order to get the maximal precision for the ``complex double'' variable, it is sufficient to make three or four iterations by the Newton method
\be
s_{n+1}=s_n + \frac{z-\mathrm{ArcTania}(s_n)}{\mathrm{ArcTania}'(s_n)}
~ ~ ~ \eL{ssTania}
where 
$\mathrm{ArcTania}'(z)\!=\!1\!+\!1/z$  
is derivative of ArcTania.
I remind that, ArcTania by equation \rf{ArcTaniaz} is elementary function.
In such a way, Tania can be evaluated quickly and precisely. The C++ implementation of function Tania with this algorithm for ``complex double'' variables is loaded as
\url{https://mizugadro.mydns.jp/t/index.php/Tania.cin}
\vskip 2mm

\JP{
Элементарные функции, упомянутые в таблице \ref{TableOfSuperfunctions}, и, в частности, Таня и АркТаня, имеют широкую область голоморфизма. Интуитивно понятно, что если суперфункция выражена несложной элементарной функцией, то, вероятно, именно она имеет физический смысл, 
а суперфункция, получаемая преобразованием \rf{tif}, есть что-то дополнительное, искусственное и менее физичное. Также, можно ожидать, что преобразованная функция будет иметь более узкую область голоморфизма; это даёт критерий  ``физичности'' решения передаточного уравнения \rf{transfereq}. Критерий простоты указан в последней, Шестой аксиоме ТОРИ, упомянутой Главе 2.
Согласно этой аксиоме, от специальных суперфункций приходится ожидать наиболее простого поведения в комплексной плоскости.
\vskip 2mm }%JP


Function Tania is simpler than function WrightOmega. If necessary, Tania can be used to evaluate WrightOmega.
\vskip 2mm 

Function Tania has simple physical meaning. It represents dependence of intensity of light on the length of its propagation in laser with
simple model of the active medium. The argument has sense of coordinate measured in units of the inverse increment of the low signal. The returned value has sense of intensity, measured in the units of saturation.
\vskip 2mm

\begin{wrapfigure}{r}{70pt}
\vskip -16pt
\sx{.7}{\begin{picture}(97,142)
\put(2,0){\ing{ValerieDoya.jpg}}
\end{picture}}
\vskip -8pt
\caption{V.Doya\cite{valerie}\label{valerie}$\!\!\!\!\!\!\!$}
\vskip -4pt
\end{wrapfigure}

In this Book, function Tania is used many times. In this chapter, Tania appears as superfunction of the special function Doya, considered in the nest section. I use the trick, mentioned in the chapter 3: First, I choose superfunction, id est, Tania, and then, I construct the transfer function for it; I call this transfer function ``Doya''.
I am grateful to Val\'erie Doya
(Figure \ref{valerie}), she kindly allowed to use her name for the function, that appeared during our collaboration at Nice in 2010.
\vskip 2mm


\section{Transfer function Doya}
\baselineskip 21pt

\begin{figure}[t]
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\put(28,-1.9){\sx{.4}{$-1$}}
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\put(3.3,9){\rot{-2}\sx{.6}{$v\!=\!-3$}\ero}
\put(66,40.6){\rot{0}\sx{.6}{$v\!=\!0$}\ero}
\put(42,40.6){\rot{60}\sx{.6}{$u\!=\!0$}\ero}
\put(59,37){\rot{86}\sx{.6}{$u\!=\!1$}\ero}
\end{picture}
}\\ \normalsize
\url{https://mizugadro.mydns.jp/t/index.php/File:Doya10map4bT60.png}\\
\url{https://mizugadro.mydns.jp/t/index.php/File:Doyam10map4bT100.png}
\vspace{-2mm}
\caption{\large
$u\!+\!\mathrm i v\!=\!\mathrm{Doya}(x\!+\!\mathrm i y)$, left, and $u\!+\!\mathrm i v\!=\!\mathrm{Doya}^{-1}(x\!+\!\mathrm i y)$, 
at right
\label{DoyaMap} \label{figDoyaMap}}
\vskip 18pt
\end{figure}

Functions Tania and ArcTania from the previous section allow to build-up the ``solvable'' transfer function. I call it Doya,
\be\!\!\!\!\!\!\!\!\!\!\!\!
T(z)=\mathrm{Doya}(z)=\mathrm{Tania}\big(1+\mathrm{ArcTania}(z)\big)
\eL{Doya}
Complex map of function Doya, and also map of its inverse function
$\mathrm{ArcDoya}=\mathrm{Doya}^{-1}$
are shown in figure
\ref{figDoyaMap}. 
At large values of the argument, each of these functions looks similar to identity function, but they have the branch points and cuts in the central part of the maps.
\vskip 2mm

While functions Tania and ArcTania are already implemented, the evaluation of function Doya is straightforward.
In addition, in vicinity of the real axis, Doya can be expressed through the known special function 
LambertW \cite{LambertW}:
\be \!\!\!\!\!\!\!\!
\mathrm{Doya}(z)=\mathrm{LambertW}\!\big(z \, \mathrm e^{z+1} \big) 
\eL{DoyaLambertW}

\begin{figure}[t]
\begin{center}
\sx{.8}{ \begin{picture}(412,412)
\put(1,9){\ing{doyaplo}}
\put(-12,394){\sx{2.8}{$y$}}
\put(-12,303){\sx{2.8}{$3$}}
\put(-12,203){\sx{2.8}{$2$}}
\put(-12,103){\sx{2.8}{$1$}}
\put(0,-9){\sx{2.5}{$0$}}
\put(100,-9){\sx{2.5}{$1$}}
\put(200,-9){\sx{2.5}{$2$}}
\put(300,-9){\sx{2.5}{$3$}}
\put(392,-7){\sx{2.6}{$x$}}
 %\put(560,214){\rot{37}\sx{4}{$y=\mathrm{Tania}(x)$}\ero}
\put(30,342){\rot{74}\sx{2.8}{$n\!=\!6$}\ero}
\put(54,342){\rot{68}\sx{2.8}{$n\!=\!5$}\ero}
\put(90,346){\rot{55}\sx{2.8}{$n\!=\!4$}\ero}
\put(146,348){\rot{53}\sx{2.8}{$n\!=\!3$}\ero}
\put(210,350){\rot{50}\sx{2.8}{$n\!=\!2$}\ero}
\put(270,340){\rot{48}\sx{2.8}{$n\!=\!1$}\ero}
\put(324,318){\rot{45}\sx{3.0}{$n\!=\!0$}\ero}
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\put(332,184){\rot{40}\sx{2.8}{$n\!=\!-2$}\ero}
\put(332,125){\rot{36}\sx{2.8}{$n\!=\!-3$}\ero}
\put(330,76){\rot{30}\sx{2.8}{$n\!=\!-4$}\ero}
\put(320,36){\rot{22}\sx{2.8}{$n\!=\!-5$}\ero}
\end{picture}}
\vskip 8pt
\url{https://mizugadro.mydns.jp/t/index.php/File:DoyaPlotT100.png}
\end{center}
\vskip -12pt 
\caption{\large $y\!=\!\mathrm{Doya}^n(x)$ по формуле \rf{DT} \iL{doyaplo}
\label{doyaite}
}
\vskip 18pt
\end{figure}

According to definition \rf{Doya}, Tania is superfunction of Doya, and ArcTania is its abelfunction.
The $n$th iterate of Doya can be written as follows:
\be
\mathrm{Doya}^n(z)=\mathrm{Tania}\Big(n+\mathrm{ArcTania}(z)\Big)
\eL{DT}
These iterates are shown in figure
\ref{doyaplo}.
The graphics represent $y\!=\!\mathrm{Doya}^n(x)$ versus $x$ for various values of the number $n$ of iterate. 
\vskip 2mm

As it is mentioned above, function Tania has simple physical sense; it describes evolution of a signal in a simple model of a uniform saturable amplifier (or absorber) at the appropriate choice of units of length and units of intensity. Similar sense can be given to function Doya, it appears as dependence to the output intensity of this amplifier on its input intensity \cite{thailand,2013or}.
\vskip 2mm

Functions Doya, Tania and ArcTania give an example, when the transfer function, the superfunction and the abelfunction can be expressed in terms of special functions, already described in the literature of 20th century. In the following section, one more example of this kind is considered.

\JP{
\begin{wrapfigure}{r}{300px}
\sx{.5}{\begin{picture}(400,240)
\put(10,-10){\ing{hillary}}
\end{picture}}
\caption{В лазерной науке слово ``сигнал'' имеет значение, отличное от бытового.
\iL{signal}}
\end{wrapfigure}

\vskip 2mm
}%JP

\section{Keller, Shoka and ArcShoka}
\baselineskip 18.8pt

\begin{figure}[t]
\begin{center}
 %\sx{1.2}{ \begin{picture}(174,165)
 \sx{2.4}{ \begin{picture}(174,172)
 \put(6,5){\ing{KellerMap}}
 \put(2,163){\sx{.7}{$y$}}
 \put(2,144){\sx{.6}{$6$}}
 \put(2,124){\sx{.6}{$4$}}
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 \put(18,147){\sx{.8}{\bf cut}}
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 \put(18,84){\sx{.8}{\bf cut}}
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 \put(18,20.8){\sx{.8}{\bf cut}}
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 \put(132,  13.4){\sx{.8}{$v\!=\!-\!7$}}
 \end{picture}}
 \url{https://mizugadro.mydns.jp/t/index.php/File:KellerMapT.png}
 \end{center}
 \vskip -18pt
 \caption{\large
 $u\!+\!v=\mathrm{Keller}(x\!+\!\mathrm i y)~$ by \rf{KellerDef}. 
  %Генератор:
 %\url{https://mizugadro.mydns.jp/t/index.php/File:ArcKellerMapT.png} и
 \iL{KellerMap} }
 \vskip 12pt
 \end{figure}

Wave packet of light (or any other quasi–classical bosons) in a uniform amplifier can be characterised with its energy or its fluence; this quantity can be denoted as ``signal''.
Roughly, fluence is energy of pulse per area of its transversal cross-section.
In analogy with the continuous-wave amplifier, the signal at the output of the amplifier can be considered as function of the input; and this dependence can be interpreted as the transfer function of the amplifier.
For the simple model of the active medium, this function can be expressed as elementary function.
Complex map of this function is down in figure \ref{KellerMap}; I call it ``Keller function'' and define it as follows:

\be
\mathrm{Keller}(z)=z+ \ln\!\Big(\mathrm e - \mathrm e^{-z}(\mathrm e-1) \Big)
\eL{KellerDef}
Complex map of this function is shown in figure
\ref{KellerMap}. \\
 %\url{https://mizugadro.mydns.jp/t/index.php/File:KellerMapT.png}
\begin{figure}[t]
  \sx{2.4}{ \begin{picture}(164,172)
\put(6,5){\ing{ArcKellerMap}}
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\put(132, 13.4){\sx{.8}{$v\!=\!-\!7$}}
\end{picture}}
\vskip 2pt
\hskip 28pt \url{https://mizugadro.mydns.jp/t/index.php/File:KellerMapT.png}
\vskip -4pt
\caption{\large  $u\!+\!v=\mathrm{ArcKeller}(x\!+\!\mathrm i y)~$ by 
\rf{ArKellerDef}. 
 %Генераторы:
 %\url{https://mizugadro.mydns.jp/t/index.php/File:ArcKellerMapT.png} и
 \iL{ArcKellerMap} }
\end{figure}
\vskip 2mm

The inverse function  $\mathrm{ArcKeller}=\mathrm{Keller}^{-1}$ can be written as follows:
\be\!\!\!\!\!\!\!\!\!\!
\mathrm{ArcKeller}(z)=z+\ln\!\left( \frac{1}{\mathrm e}+\frac{\mathrm e \!-\!1}{\mathrm e}\, \mathrm e^{-z} \right)
\eL{ArcKellerDef}
Complex map of function ArcKeller is shown in Figure 
\ref{ArcKellerMap}.
Maps of functions Keller and ArcKeller look similar; the following relation takes place:
\be
\mathrm{ArcKeller}(z)=\mathrm{Keller}(z-\mathrm i \pi -1)-1+\mathrm i \pi
\label{ArKellerDef}
\ee
\vskip 2mm

\begin{wrapfigure}{r}{60pt}
\sx{.33}{
\begin{picture}(188,200)
\put(0,0){\ing{Keller62241866.png}}
\end{picture}}
\vskip -4pt
\caption{~ U.Keller \label{ursula}}
\vskip 14pt
\end{wrapfigure}

In publications \cite{keller1,keller2}  by Ursula Keller (fig.\ref{ursula}), another representation is used,
\be \!\!\!\!\!
\mathrm{Keller}(z)= \ln\!\Big( 1+ \mathrm e \, (\mathrm e^z -1) \Big) ~ ~
\iL{KellerLit} ~ ~ ~ ~ ~
\ee
This representation is equivalent of expression \rf{KellerDef}, while $|\Im(z)|\!<\!\pi$.
In particular, representation \rf{KellerLit} can be used instead of  \rf{KellerDef} for real positive values of $z$.

\begin{wrapfigure}{r}{220px}
\sx{.7}{ \begin{picture}(300,380)
\put(-4,-4){\ing{KellerDoya}}
\put(6,384){\sx{2.6}{$y$}} 
\put(5,290){\sx{2.6}{$3$}} 
\put(5,190){\sx{2.6}{$2$}} 
\put(5, 90){\sx{2.6}{$1$}} 
\put(92,4){\sx{2.6}{$1$}}
\put(192,4){\sx{2.6}{$2$}} 
\put(280,4){\sx{2.6}{$x$}} 
\put(150,250){\rot{48}\sx{2.4}{$y\!=\!\mathrm{Keller}(x)$}\ero}
\put(180,210){\rot{47}\sx{2.4}{$y\!=\!\mathrm{Doya}(x)$}\ero}
\put(320,0){\rot{90}\url{https://mizugadro.mydns.jp/t/index.php/File:KellerDoyaT.png}\ero}
\end{picture}}
\vskip -6pt
\caption{\large Comparison of functions  Keller and Doya by \rf{KellerDef} and \rf{Doya}
\label{Keller}}
\vskip -6pt
\end{wrapfigure}
\vskip 2mm

Function Keller of real argument is compared to function Doya in figure \ref{Keller}.
These functions look similar.
\vskip 2mm


Description of pulses usually is more difficult, than consideration of the stationary regime; for pulses, there is additional parameter, time.
However, for the simple model above, the pulsed regime is simpler, than the steady-state case, because, for the pulses, the transfer function, and the superfunction can be expressed as elementary functions. In such a way, function Shoka is simpler than function Tania, and function Keller is simpler than function Doya.
\vskip 2mm

\begin{figure}[t]
\begin{center}
\sx{.9}{ \begin{picture}(410,412)
\put(1,9){\ing{kellerplo}}
\put(-12,394){\sx{2.8}{$y$}}
\put(-12,303){\sx{2.8}{$3$}}
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\put(200,-9){\sx{2.5}{$2$}}
\put(300,-9){\sx{2.5}{$3$}}
\put(392,-7){\sx{2.6}{$x$}}
 %\put(560,214){\rot{37}\sx{4}{$y=\mathrm{Tania}(x)$}\ero}
\put(28,344){\rot{76}\sx{2.7}{$n\!=\!5$}\ero}
\put(52,346){\rot{66}\sx{2.8}{$n\!=\!4$}\ero}
\put(100,348){\rot{55}\sx{2.8}{$n\!=\!3$}\ero}
\put(172,350){\rot{50}\sx{2.8}{$n\!=\!2$}\ero}
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\put(324,318){\rot{45}\sx{3.0}{$n\!=\!0$}\ero}
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\put(316,20){\rot{7}\sx{2.8}{$n\!=\!-5$}\ero}
\end{picture} }
\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:KellerPlotT.png}
\end{center}
\vskip -14pt
\caption{\large $y\!=\!\mathrm{Keller}^n(x)$ .
\iL{kellerite}
\label{KellerPlot}}
\vskip 12pt
\end{figure}


Iterates of function Keller are shown in figure
\ref{KellerPlot}.
They look similar to iterates of function Doya shown in figure 
\ref{doyaplo}. 
By general formula \ref{Tzx}, the iterates of the Keller function can be expressed through its superfunction 
$F\!=\!\mathrm{Shoka}$
and abelfunction
$G\!=\!\mathrm{ArcShoka}$; these functions can be expressed as follows:
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
 \mathrm{Shoka}(z)=z+\ln\!\Big( \mathrm e^{-z} +\mathrm e -1\Big)
\eL{Shoka}
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{ArcShoka}(z)= z + \ln \!\left(\! \frac{1\!-\!\mathrm e^{-z}}{\mathrm e \!-\!1} \right)\!\!
\eL{ArcShoka}

\begin{figure}[h]
\begin{center}
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\put(6,5){\ing{ShokaMap}}
\put(2,163){\sx{.7}{$y$}}
\put(2,144){\sx{.6}{$6$}}
\put(2,124){\sx{.6}{$4$}}
\put(2,104){\sx{.6}{$2$}}
\put(3,116){\sx{.6}{$\pi$}}
\put(18,161.4){\sx{.8}{$u\!=\!0$}}
\put(11,150){\sx{.8}{$v\!\approx\!2\pi$}}
\put(18,131.2){\sx{.8}{$u\!=\!0$}}
\put(18,115.6){\sx{.8}{\bf cut}}
\put(18,100){\sx{.8}{$u\!=\!0$}}
\put(2, 84){\sx{.6}{$0$}}
\put(18, 84){\sx{.8}{$v\!=\!0$}}
\put(18,68){\sx{.8}{$u\!=\!0$}}
\put(-2,64){\sx{.6}{$-2$}}
\put(-2,53){\sx{.6}{$-\pi$}}
\put(18,52.4){\sx{.8}{\bf cut}}
\put(18,36.5){\sx{.8}{$u\!=\!0$}}
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\put(132, 53.4){\sx{.8}{$v\!=\!-\!3$}}
\put(132, 43.4){\sx{.8}{$v\!=\!-\!4$}}
\put(132, 33.4){\sx{.8}{$v\!=\!-\!5$}}
\put(132, 23.4){\sx{.8}{$v\!=\!-\!6$}}
\put(132, 13.4){\sx{.8}{$v\!=\!-\!7$}}
\end{picture}}
\url{https://mizugadro.mydns.jp/t/index.php/File:ShokaMapT.png}
\end{center}
\vskip -18pt
\caption{$u\!+\!\mathrm i v\!=\!\mathrm{Shoka}(x\!+\!\mathrm i y)$
 \iL{ShokaMap}}
\vskip 12pt
\end{figure}

\begin{figure}[t]
\begin{center}
\sx{2.4}{ \begin{picture}(174,174)
\put(6,5){\ing{ArcShokaMap}}
\put(2,163){\sx{.7}{$y$}}
\put(2,144){\sx{.6}{$6$}}
\put(2,124){\sx{.6}{$4$}}
\put(2,104){\sx{.6}{$2$}}
\put(18,147){\sx{.8}{\bf cut}}
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\put(18,84){\sx{.8}{\bf cut}}
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\put(18,20.8){\sx{.8}{\bf cut}}
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\put( 22,0){\sx{.6}{$-6$}}
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\put( 62,0){\sx{.6}{$-2$}}
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\put(132, 94.2){\sx{.8}{$v\!=\!1$}}
\put(132, 84){\sx{.8}{$v\!=\!0$}}
\put(132, 73.9){\sx{.8}{$v\!=\!-\!1$}}
\put(132, 63.4){\sx{.8}{$v\!=\!-\!2$}}
\put(132, 53.4){\sx{.8}{$v\!=\!-\!3$}}
\put(132, 43.4){\sx{.8}{$v\!=\!-\!4$}}
\put(132, 33.4){\sx{.8}{$v\!=\!-\!5$}}
\put(132, 23.4){\sx{.8}{$v\!=\!-\!6$}}
\put(132, 13.4){\sx{.8}{$v\!=\!-\!7$}}
\end{picture}}
\url{https://mizugadro.mydns.jp/t/index.php/File:ArcShokaMapT.png}
\end{center}
\vskip -16pt
\caption{\large $u\!+\!\mathrm i v\!=\!\mathrm{ArcShoka}(x\!+\!\mathrm i y)$. 
 \iL{ArcShokaMap}}
\vskip 4pt 
\end{figure}

Complex maps of functions Shoka and ArcShoka are shown in figures \ref{ShokaMap} and \ref{ArcShokaMap}. 
These maps
look similar to those for functions Tania and ArcTania shown in figures \ref{TaniaMap} and \ref{ArcTaniaMap}. 
\vskip 2mm

For real values of the argument, functions Tania and Shoka are compared in figure 
\ref{shokataniaplot} mentioned in the preamble of this chapter. 
Both functions in the left hand side of the graphic have the exponential growth with increment unity;
both pass thorough point (0,1) and both grow almost linearly in the right hand side of the graphics.
\vskip 2mm

Complex maps of functions Shoka and ArcShoka in figures \ref{ShokaMap} and \ref{ArcShokaMap} look similar.
One of them can be obtained from another with constant displacement of the argument and addition of some constant to its value. This can be expressed with relation
\be  \!\!\!\!\!
\mathrm{ArcShoka}(z)=\mathrm{Shoka}\big(z-\mathrm i \pi -\ln(\mathrm e\!-\!1)\big)-\ln(\mathrm e\!-\!1)+\mathrm i \pi
\label{ashsh}\ee
In vicinity of the positive part of the real axis, the map of function Shoka in figure
\ref{ShokaMap}
looks similar to the map of function Tania at Figure 
\ref{figTaniaMap}. %для функции Tania,
However, I cannot suggest any simple expression of function Tania through function Arctania.
I know no analogy of formula \rf{ashsh} for functions Tania and ArcTania.
\vskip 2mm

There are also qualitative differences between Tania and Shoka.
Tania had only two branch points, and accordingly, two cut lines.
Shoka has countable set of branch points and cuts.
\vskip 2mm

All the six functions Tania, ArcTania, Doya, Shoka, ArcShoka and Keller look similar to the linear function at the large values of the real part of the argument. In the right hand side of the complex maps in figures 
\ref{TaniaMap}, 
\ref{ArcTaniaMap}, 
\ref{DoyaMap},
\ref{ShokaMap}
\ref{ArcShokaMap},
\ref{KellerMap},
the structure of levels of constant real part and levels of constant imaginary part form 
grid of lines, almost parallel to the real or imaginary axes.
\vskip 2mm

\begin{wrapfigure}{r}{120px}
\sx{5}{\begin{picture}(16,28)
\put(1,-1){\ing{doya500}}
\put(25,-7){\rot{90}\sx{.116}{\url{https://mizugadro.mydns.jp/t/index.php/File:Doya500.png}}\ero}
\end{picture}}
\vskip -4pt
\caption{ 
Eureka! 
\label{eureka0}}
\vskip -12pt
\end{wrapfigure}
In order to catch (or to remember) the properties of function Doya, the 
left hand side part of figure \ref{DoyaMap} is repeated in figure
\ref{eureka0}. 
The map is shown with $u\!+\!\mathrm i v \!=\!\mathrm{Doya}(x\!+\!\mathrm i y)$,
but the picture is rotated for $90^{\rm o}$ in the positive direction (counter-clock wise).
The thick lines show levels
$u\!=\!-0.4$,  
$v\!=\!\pm 1.2$~, 
$v\!=\!\pm 1.4$~
The map looks as a draw of a Researcher, who has found something extraordinary in a big Book.
Perhaps, that Book is about superfunctions.
%Generator of this image is loaded as
%\url{https://mizugadro.mydns.jp/t/index.php/File:Doya500.png}~. 
The Reader is invited to check that values of the 3 parameters, indicated above, provide the structure, shown in figure
\ref{eureka0}. The range of the almost linear transfer appears above the ``head'' of the contour of the  ``human'' in the figure.
\vskip 2mm

Similarity of the transfer functions Doya and Keller and, accordingly, similarity of their superfunctions Tania and Shoka indicate, that for analysis of the nonlinear media, these functions should be measured with several significant figures; over-vice, the measurement will not allow to make choice between divergent models.  At the convenient measurement 
of the nonlinear response of the medium, variation of the intensity in a sample should be small:
over-vice, it is difficult to guess, namely which intensity does the gain or absorption correspond to. At small variation of the intensity, the precision of its measurement is poor. This difficulty can be avoided with superfunctions. The transfer function of the optically-thick sample should be measured, and then, the superfunction, id est, the evolution of intensity inside the sample can be reconstructed.
In such a way, the number of parameters of the model, can be reduced:
one of parameters, namely, the length of the amplifier, can be excluded from the model.


\vskip 2mm
My particular
interest in superfunctions is related to the ability of the precise measurement of the gain (or absorption) 
in the nonlinear medium versus intensity. 
The application may refer to investigation of limits of validity of the commonly-used model of the Yb-doped crystals, glasses and ceramics. The new effects are expected to appear at the edge of the limit of applicability.
One of such effects is described in the article about switching of emissivity of Yb-doped samples, or the ``Bisson effect'' \cite{BissonEffect},
but namely in that case, the main mechanism of the phenomenon seems to have thermal origin: the sample warms, and this warming enhances the absorption (and heating), leading to the avalanche behaviour, that is difficult to interpret in terms of the transfer function of a single variable, considered in this Book \footnote{The name ``Bisson effect'' or `bison effect'' refers to the analogy of the avalanche of the tribe of running bison, who awake, mock and force to run the new and new animals. However, the analogy is limited, because the electrons, that were expected to be popped up into the conduction band, are neither bison, nor even bosons, but fermions.} .

\section{Overview}

The six functions, (Tania, Shoka, ArcTania, ArcShoka, Doya and Keller) are defined in this chapter.
These functions show the realistic examples, the physically-meaningful transfer functions and the corresponding meaningful superfunctions are expressed in terms of elementary functions. These example will be used in this book later, to illustrate more general methods of construction and evaluation of superfunctions.
\vskip 2mm

My main claim is that I can construct the superfunction, abelfunction and, therefore, the non-integer iterates of any growing real-holomorphic function
\footnote{With the same methods, the non-real, but still holomorphic functions also can be treated; such an example is considered in chapter 18.}.
Even more, everybody, after to read this Book, also can do the same. 
\vskip 2mm

In the following chapters, I consider methods, that can be used for various transfer functions, even if the superfunction cannot be expressed through the special functions known since century 20. One of these cases is considered in the next chapter.


\chapter{~ Regular iteration \label{Regular iteration}}
\baselineskip 18.4pt

Regular iteration is way to construct iterates of the transfer function, that are regular in vicinity of its fixed point.
The transfer function is supposed to be holomorphic. 
In the most of examples, considered in this Book, the transfer function is assumed to be also real-holomorphic. In addition, in this chapter, I assume, that the fixed point is real.
\footnote{The same regular iteration can be applied also for the case of non-real fixed point, but 
the resulting iterates of the transfer function are not real-holomorphic. The real-holomorphic functions often are easier to interpret (and to apply in Physics), than the more general complex case. 
This Book is planned as applied, so, the most of examples here refer to the real-holomorphic functions.}
\vskip 2mm

This Book deals with solutions $F$ of the transfer equation
\rf{transfereq}; I repeat it here
\be
F(z\!+\!1)=T\big(F(z)\big)
\nonumber \ee
Assume, that the transfer function $T$ is real-holomorphic, and its fixed point $L$  is real, id est,
\be
T(L)=L~~, ~ ~ ~ 
L=L^*
\ee
In addition, I assume, that  $T'(L)\!>\!0$, this case is easier to interpret
\footnote{%Предположение $T'(L)\!>\!0$  естественно. Случай $T'(L)\!<\!0$ трудно интерпретировать в терминах вещественных значений нецелой итерации, так как ${T^n}^\prime(L)$ должна стремиться  к единце при $n\!\rightarrow\!0$. Читатели могут рассмотреть также случай, когда $T'(L)\!=\!0$.
Assumption $T'(L)\!>\!0$ is natural. Case $T'(L)\!<\!0$ 
is difficult to interpret in terms of real valued non-integer iterates, because, 
${T^n}^\prime(L)$ should approach unity at $n\!\rightarrow\!0$, but, for realistic cases, should avoid zero.
Practically, this means that the non-integer iterates are complex, not real, as in the case $T'(L)\!>\!0$
 } .

For this case, In this chapter, the specific iterates of the transfer function are constructed, that can be expanded into  convergent Taylor series at least in vicinity of the fixed point $L$. In this sense, they are regular. On the other hand, the iterates are constructed with definite iteration procedure, described below, in a regular straightforward way. Hence, there are at least two independent reasons to use name ``regular iteration'' for the procedure below. 
\section{General formula}
\baselineskip 18pt

Assume that $L$ is fixed point of the transfer function $T$, id est, $T(L)=L$. For the transfer equation  \rf{transfereq}, I search for the asymptotic solution
$F$ in the following form: 
\be
\tilde{F}(z)=L+\varepsilon+a_2 \varepsilon^2+ a_2 \varepsilon^3 + ..
\eL{Lea2general}
where
\be
\varepsilon=\exp(kz)
\eL{Lea2}
Here, $k$ is constant, that bas sense of the increment, and $a$ refer to some set of real constants, that do not depend on $z$.
As it is indicated in the Preamble of this chapter, I keep in mind the case when 
$T(z^*)=T(z)^*$  и 
$F(z^*)=F(z)^*$; however, in principle, the expansion \rf{Lea2} can be used for more complicated case too (and in this case the coefficients $a$ should be complex).
\vskip 2mm

The consideration could be generalised, adding new terms in to the right hand side equation  \rf{Lea2general}, for example, terms, that are not polynomial with respect to  $\varepsilon$.
Following my wishes about the popularity
\footnote{Here, the term ``popularity'' may have each of the two its meanings},
below I consider the simplest (but still non-trivial) case.
\vskip 2mm

Substituting $F \rightarrow \tilde F$ into the transfer equation  \rf{transfereq}, which is 
\be
F(z+1)=T(F(z))
\nonumber
\ee
in the left hand side I get the following
\be
\tilde F(z\!+\!1)=L+\mathrm e^k \varepsilon+ a_2 \mathrm e^{2k} \varepsilon^2 + a_3 \mathrm e^{3k} \varepsilon^3 + ..
\eL{iteraz1}
and the right hand side gives
\be
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
T(\tilde F(z))&=&L  + T' \cdot \varepsilon+ T' \cdot a_2 \varepsilon^2 + T' \cdot a_3 \mathrm \varepsilon^3 + .. \nonumber \\
&&+ \frac{T''}{2} (\varepsilon\!+\! a_2 \varepsilon^2 \!+\! ..)^2 + \frac{T'''}{6} (\varepsilon\!+\! ..)^3 + ..
\eL{i2}
where $T' =T'(L)$, $T'' =T''(L)$, $T''' =T'''(L)$, .. are derivatives of the transfer function $T$ at the fixed point $L$.
\vskip 2mm

coefficients at the same power of $\varepsilon$ in expression \rf{iteraz1} and in expression \rf{i2} should be equal.
This gives the equations for increment $k$  and coefficients $a$:
\be
\mathrm e^k &=& T' \\
\mathrm e^{2k} a_2 &=& T' a_2 + T''/2 \\
\mathrm e^{3k} a_3 &=& T' a_3 + T''a_2 + T'''/6 \\
.. \nonumber
\ee
The chain of these equations 
determines
\be
k&=& \ln(T')  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~\iL{k38}\\
a_2&=& \frac{T''/2}{(T'-1)T'} ~ ~ ~ ~ ~ ~ ~ ~ \iL{a2.39}\\
a_3&=& \frac{T''a_2+T'''/6}{((T')^2-1)T'} ~ ~ ~ ~ \iL{a2.40}\\
.. \nonumber
\ee
Typically, for a simple special function $T$, the Mathematica, the Maple or any similar software allow to evaluate tens of coefficients $a$ in real time.\footnote{My experience indicates, that, for applications, the evaluation of all quantities in real time is so important, as the real holomorphism of the functions. In such a way, I try to be realistic, dealing with real quantities in the real time. However, even for the real-holomorphic superfunctions, their behaviour at complex values of argument is important and should be considered; but even in this case I try to keep algorithms short and fast, to make evaluations in the real time.}
The truncated series in representation \rf{Lea2general} 
provides good approcimation for $F$ at $\varepsilon\ll 1$.
For positive values $k$, this corresponds to the large negative values of $\Re(z)$. 
For other values, the approximation can be improved with 
\be
F(z)\approx T^n \Big( \tilde F (z-n) \Big)
\eL{regi}
for the large enough positive values of $n$ at positive $k$, and
for the large enough negative values of $n$ at negative $k$.
Case
$k\!=\!0$ 
is qualified as exotic and is considered below in the special chapter about exotic iterates.
\vskip 2mm

In may cases, representation 
\rf{regi}
allows to evaluate the superfunction with the required precision.
In particular, this allows to evaluate the super exponentials to base 
$b<\exp(1/\mathrm e)$
\cite{sqrt2},
superfactorial \cite{factorial}, 
and holomorphic extension of the logistic sequence \cite{logistic},
and even the holomorphic extension of the Collatz subsequence \cite{collatz}.
Some of these examples are considered in the following chapters.
\vskip 2mm

I call this case ``regular iteration'', as the iteration by 
\rf{Tzx}, 
id est,
$T^n(z)=F\big(n\!+\!F^{-1}(z)\big)$ 
is holomorphic (``regular'') function in vicinity of the stationary point 
$z\!=\!L$.
For real fixed point  $L$, 
I expect, namely regular iteration corresponds to the physically-meaningful solution $F$ of the transfer equation
\rf{transfereq}. 
This statement can be qualified as 
conjecture.
Following the TORI axioms, I tried to negate, refute this conjecture, but I got confirmations instead.
These confirmations, examples form the significant part of this Book.

\JP{
Neither colleagues, nor I could construct any case, when the meaningful regular iteration 
of the transfer function
for the case 
$T(L)\!=\!L$, $T'(L)\!>\!0$, $L\!=\!L^*$
cannot be expressed through the regular iteration procedure above.
In this sense, I consider as scientific fact, that namely the regular iteration provides the physically-meaningful solution for this case.
}
\section{Exact solution}
\baselineskip 18pt

The approximate equality in expression \rf{regi}
should not make an impression, that the regular iteration gives only approximation of the superfunction.\footnote{
One of my coauthors, until now, believes, that even $\pi$ is approximate number. This can be considered as a kind of a mental illness, as well as some kind of religion.. However, if for any given $\alpha>0$, the error of evaluation of some quantity can be done smaller, than $\alpha$,
then the quantity is considered as exact. Then, we may consider the CPU time and number of flops required to get some given precision of the approximate evaluation of superfunction $F$ in \rf{regi}. }
For the case of $\Re(k)\!>\!0$ (or, in particular, for $k\!>\!0$), the exact superfunction can be expressed with limit
\be
F(z)=\lim_{n\rightarrow \infty} \tilde T^n\big(F_m(z-n)\big)
\eL{regitexa}
where 
\be
F_m=L+\varepsilon + \sum_{\ell=2}^{m} a_n \varepsilon^n
\ee
and $k$, $\varepsilon$ and coefficients $a$ are defined in the previous section; while  $m\!\ge\!2$ is some integer number.
Due to the asymptotic properties of the solution $\tilde F$, the limit in expression \rf{regitexa} does not depend on $m$.
However, the convergence is faster for larger $m$.
Similar limit can be written for the case $\Re(k)\!<\!0$.
\vskip 2mm

The series \rf{Lea2} can be inverted, giving the asymptotic approximation $\tilde G$ of the abelfunction $G\!=\!F^{-1}$.
Then, the exact abelfunction can be expressed with
\be
G(z)= \lim_{n\rightarrow \infty} \ln \left(\frac{1}{k} \tilde G_m(T^{-n}(z)) \right)+ n
\ee
where $\tilde G_m$ is some truncation of the asymptotic expansion for the abelfunction.
The iterate of the transfer function, $T^n(z)=F(n+G(z))$ happen to be regular in vicinity of $z\!=\!L$.
In the following chapters, this statement is verified for many specific realisations of the transfer function $T$.
\JP{
\section{Induction, deduction and scientific facts}
\baselineskip 20.2pt

Этот раздел пишется для тех математиков, которые считают дедукцию царицей доказательств, а все проверки с частными случаями считают упражнениями ниже своего достоинства. Ниже предлагается философское (и несколько лирическое) отступление от генеральной линии изложения. Речь идет об интерпретации термина ``научный факт'', использованного в предыдущей секции.
\vskip 2mm

Мне приходилось иметь дело с математиками, которые в качестве подтверждения концепции признают только её строгий математический вывод. Я был склонен принять такую точку зрения, но меня отвратило одно неумолимое эмпирическое наблюдение:
Результаты, полученные в результате ``чистой'', строгой дедукции, чаще оказываются ошибочными, чем ``догадки'' (конжекции), сделанные на основе анализа частных случаев и подтвержденные попытками опровергнуть результат.
\vskip 2mm

Об одном случае, иллюстрирующем упомянутую выше закономерность, рассказал Юрий Широков. 
Он работал с одной Аспиранткой, и предложил ей рассчитать некоторую физическую зависимость. Аспирантка долго мучилась с выводом, но провела рассчет и принесла формулу. Юрий посмотрел и сказал: ``У вас ошибка''. 
Аспирантка
повторила свои выкладки, и стала настаивать на своем результате. 
Она требовала, чтобы Юрий проверил её рассчеты.
Юрий проверять рассчеты не хотел. Аспирантка почти насильно заставила Юрия проследить её дедукцию. Они вместе проверяли строчку за строчкой, Юрий читал и говорил ``Угу'',
но в последней строчке, когда потребовалось применить формулу из справочника, Юрий сказал: ``Здесь ошибка''.
Аспирантка открыла Градштейна-Рыжика и показала формулу. Юрий посмотрел на формулу и повторил, что формула неверна.
В той формуле было достаточно положить параметр равным нулю, чтобы увидеть, что равенство нарушено.
Аспирантка обиделась: в списке опечаток, который добавлялся к каждому новому переизданию,
про пресловутую формулу ничего не было. Юрий сказал:``Значит, будет''. Но он не смог убедить Аспирантку, что если она пользуется неверной формулой, и не считает нужным сделать проверку, хотя бы для 
частных случаев, то это её вина.
\vskip 3mm

Я тоже не без греха. В рассчет усиления при выжигании пространственных дыр закралась ошибка \cite{thin,thine}. Общая формула не использовалась для оценок, и ошибка не была выявлена сразу. С тех пор я не надеюсь, что соавторы выловят ошибки, и стараюсь не предлагать рассчеты, которые не используются для построения рисунков и формулировки основного ``мессаджа'' (абстракта и заключения) статьи. Это соответствует аксиоме 6: надо рассматривать самые простые построения, которые всё ещё приводят к нетривиальной концепции.
\vskip 3mm
   
Иногда мне не удавалось убедить коллег, указывая на противоречия их концепций; приходилось продираться сквозь выкладки и публиковать соответствующий ерратум. Некоторые такие примеры упомянуты в УФН \cite{ufn} и J.Mod.Phys.\cite{2013jmp}. 
На основе этих наблюдений, я считаю, что применение аксиом ТОРИ более эффективно и надежно, чем громоздкая дедукция, претендующая на математическую строгость. В идеале, конечно, желательно сочетать и математическую строгость, и 
всяческие проверки частных случаев, асимптотик и численных рассчетов.
\vskip 3mm

Всего один раз я столкнулся со случаем, когда четырнадцати значащих цифр оказалось недостаточно для численного опровержения гипотезы о том, что две голоморфные функции тождественно совпадают \cite{sqrt2}. Я был почти уверен в том, что эта разница вылезет при достаточно аккуратных рассчетах, так как две голоморфных функции не могут совпадать на вещественной оси, если они отличаются хоть где-то в связной области их голоморфизма. 
Для вещественнoго аргумента, разница между значениями функций проявляется в 25-м десятичном знаке.
Этот случай упомянут ниже в главе 16.
\vskip 3mm

В физической математике, строгая дедукция не является царицей доказательств; примерно так же, как в правовом обществе признание обвиняемого не является царицей доказательств его вины. Если утверждение удается доказать строго, то есть вывести на основе четкой математической выкладки - хорошо, но это не избавляет исследователя от необходимости проверять частные случаи и пытаться 
построить хотя бы один опровергающий пример. 
В следующей секции я пытаюсь построить такой опровергающий пример.
}%JP
\JP{
Если же простая гипотеза предсказывает сразу ворох специальных и нетривиальных явлений, и не удается найти ни одного контрпримера
--
я считаю её
научным фактом,
и именно такие факты я предлагаю в этой книге. 
Если Вам удастся опровергнуть хотя бы одну из моих конжекций, вы можете написать об этом научную статью, и предложить более правильную формулу, которая, в частности, будет объяснять, почему во всех рассмотренных мной случаях моя формула хорошо работает. Я стараюсь помочь Вам писать такую статью, и в этой Книге я собираю формулы и алгоритмы, которые можно использовать при написании опровергающей статьи. 
\vskip 2mm

В соответствии с Третьей аксиомой ТОРИ, мои утверждения опровержимы; достаточно найти, построить хотя бы один случай, когда формула приводит к противоречию, чтобы опровергнуть её. Все нетривиальные примеры, в которых формула хорошо работает и дает ожидаемый результат, могут рассматриваться как неудачные попытки её опровержения. В следующей секции я пытаюсь опровергнуть метод регулярной итерации, предлагаемые в этой главе, используя для этого задачу с известным ответом.
}%JP
\vskip 2mm

Before to apply the regular iteration to the new transfer functions, for which the superfunctions cannot be easy expressed through the superfunctions, known in century 20, it worth to check the method for some easy function, for which the answer is known. First, I use the method of regular iteration to construct the superfunction for the transfer function Doya, considered in the previous chapter. I do it, as if I would not know that its superfunction is Tania. This regular iteration is considered in the next section, and compared to the exact solution Tania.

\section{Example with known solution: again Doya}
\baselineskip 19pt

\begin{wrapfigure}{r}{180px}
\normalsize
\sx{.8}{
\hskip 4pt  \begin{picture}(200,298)
\put(0,0){\ing{doyaplot1}}
\put(-7,300){\sx{1.6}{$y$}}
\put(-7,200){\sx{1.6}{$2$}}
\put(-7,100){\sx{1.6}{$1$}}
\put(0,-11){\sx{1.6}{0}}
\put(100,-11){\sx{1.6}{1}}
\put(198,-11){\sx{1.6}{$x$}}
\put( 38,110){\rot{70}{\sx{1.7}{$y=\mathrm e\, x$}}\ero}
\put( 58,110){\rot{53}{\sx{1.7}{$y=\mathrm{Doya}(x)$}}\ero}
\put( 24,30){\sx{1.6}{$y=\mathrm e\, z - \mathrm e (\mathrm e\!-\!1) x^2$}}
\put(220,-12){\rot{90}\url{https://mizugadro.mydns.jp/t/index.php/File:Doya1T.jpg}\ero}
\end{picture}} 
\caption{%$T\!=\!\mathrm{Doya}$ 
\baselineskip 13pt
Function Doya and\\ %two its approximations
its polynomial approximations
\label{f0fig}
}%
\vskip -12pt
\end{wrapfigure}

In this section, I show, how does the regular iteration work.
I use the example with transfer function Doya, described in the previous chapter.
For real value of argument, graphic of this function is shown in figure 
\ref{f0fig} 
with thick curve.
\vskip 2mm

For transfer function $T\!=\!\mathrm{Doya}$, the superfunction is known, it is function 
$F\!=\!\mathrm{Tania}$, and the corresponding abelfunction is $G\!=\!\mathrm{ArcTania}$.
Properties of these functions are described in publications 
\cite{2013or,tania,doya} and in the previous chapter.
But, in order to check the regular iteration, assume for a minute, that we do not know the analytic expression for the superfunction, and want to construct it with the regular iterations described above. After to construct it, we may compare the result with the special function Tania.
\vskip 2mm

Let the transfer function, at least in some vicinity of the real values of the argument, can be expressed with formula  \rf{DoyaLambertW}, I repeat it here:
\be 
T(z)=\mathrm{Doya}(z)=\mathrm{LambertW}(1\!+\!z\, \mathrm e^z)
\eL{doya1}
Properties of function LambertW are known
\cite{LambertW,LambertWm,LambertWs,LambertWw}.
This function can be used even without to refer to function Tania; this emulates situation, when the superfunction $F$ is not known.
\vskip 2mm

Graphic of function Doya is shown in figure \ref{f0fig} with thick curve.
This transfer function describes some idealised amplifier with saturation, neglecting the spontaneous emission (and many other physical effects).
In this case, the fixed point $L\!=\!0$ should be considered;
$T(0)\!=\!0$. 
At small intensity at the input, the amplification coefficient is $\mathrm e$, but it saturates at the intensity of order of unity.
For small argument, the transfer function can be expanded as follows:
\be \!\!\!\!\!\!\!\!\!\!\!
T(z)=\mathrm{Doya}(z)=\mathrm e\, z - \mathrm e\, (\mathrm e \!-\!1) z^2 + O(z^2) ~ ~
\eL{DoyaExpa}
In this case, $T'(0)\!=\!\mathrm e$ and $T''(0)\!=\!- 2 \mathrm e (\mathrm e \!-\!1)$. 
The corresponding linear and quadratic approximations are shown in figure
\ref{f0fig} with thin lines.
\vskip 2mm

For regular iteration of transfer function Doya, using formulas \rf{k38} and \rf{a2.39}, I found $k\!=\!1$ and  $a_2\!=\!-1$.
In such a way, the primary approximation 
with singe term is
\be
\tilde F(z)=\exp(z)
\eL{refitF1}
and the primary approximation with two terms is
\be
\tilde F(z)=\exp(z)-\exp(2z)
\eL{refitF2}
These two primary approximations are shown in figure
\ref{regufig} 
with the highest and lowest curves.
At the same figure, the four iterations of these primary approximations by
(\ref{regi}) are shown for $n\!=\!0 .. 4$.
These approximations approach the exact solution
\begin{figure}
\begin{center}
\sx{.52}{
\begin{picture}(800,408)
\put(0,0){\ing{simudoya}}
\put(-12,297){\sx{2.2}{$3$}}
\put(-12,197){\sx{2.2}{$2$}}
\put(-12, 97){\sx{2.2}{$1$}}
\put(-12, -3){\sx{2.2}{$0$}}
\put( 88,-16){\sx{2.2}{$-2$}}
\put(188,-16){\sx{2.2}{$-1$}}
\put(299,-16){\sx{2.2}{$0$}}
\put(400,-16){\sx{2.2}{$1$}}
\put(500,-16){\sx{2.2}{$2$}}
\put(600,-16){\sx{2.2}{$3$}}
\put(700,-16){\sx{2.2}{$4$}}
\put(797,-16){\sx{2.2}{$x$}}
\put(366,206){\rot{65}\sx{1.9}{$\exp(x)$}\ero}
\put(436,206){\rot{66}\sx{1.9}{$T(\exp(x\!-\!1))$}\ero}
\put(530,266){\rot{66}\sx{1.9}{$T^2(\exp(x\!-\!2))$}\ero}
\put(580,266){\rot{60}\sx{1.9}{$T^3(\exp(x\!-\!3))$}\ero}
\put(620,266){\rot{55}\sx{1.9}{$T^4(\exp(x\!-\!4))$}\ero}
\put(733,322){\rot{41}\sx{2}{$F(x)$}\ero}
\put(534,160){\sx{1.85}{$T^4(\exp(x\!-\!4)\!-\!\exp(2(x\!-\!4)))$}}
\put(508,116){\sx{1.86}{$T^3(\exp(x\!-\!3)-\exp(2(x\!-\!3)))$}}
\put(418,64){\sx{1.86}{$T^2(\exp(x\!-\!2)-\exp(2(x\!-\!2)))$}}
\put(280,35){\sx{1.86}{$T(\exp(x\!-\!1)-\exp(2(x\!-\!1)))$}}
\put(160,11){\sx{1.86}{$\exp(x)-\exp(2x))$}}
\end{picture}}
\vskip 10pt
\url{https://mizugadro.mydns.jp/t/index.php/File:SimudoyaTb.png}
\end{center}
\vskip -12pt
\caption{Superfunction $F\!=\!\mathrm{Tania}$ and its approximations $\tilde F$ by (\ref{regi}) for $n\!=\!0 .. 4$}
\label{regufig}
\vskip 8pt
\end{figure}
\be
F(z)=\mathrm{Tania}(z\!-\!1)=\mathrm{WrightOmega}(z)
\ee
This function $F$ is shown in figure  \ref{regufig} with thick line. It remains between approximations
obtained with iterate from the primary approximation $\tilde F$ with single term, id est, \rf{refitF1} 
and those with tho terms by \rf{refitF2}.
\vskip 6pt

Example with the transfer function Doya shows the efficiency of the regular iteration.
In the following chapters, the regular iteration is used also for other transfer functions; I mean, for the cases, when the superfunction
cannot be easy expressed in terms of the special functions, known in century 20.
\section{Schr\"oder equation}
\baselineskip 17.4pt
\begin{wrapfigure}{r}{60px}
\sx{.22}{\begin{picture}(100,138)
\put(-2,-10){\ing{eschroeder}}
\put(-2,-140){\sx{4.2}{E.Schr\"oder}}
\end{picture}}
\vskip -6pt
\caption{\label{eschroeder}}
\end{wrapfigure}

For the regular iteration, instead of superfunctions (presented in this Book),
another formalism can be used, namely, the formalism of the Schroeder (Schr\"oder) functions; they are called after Ernest Schr\"oder, shown in figure 
\ref{eschroeder}, and described many years ago
\cite{szekeres,shapiro}. %(не представленный в этой Книге). 
\vskip 6pt
Let transfer function $T$ be real-holomorphic, and let $L\!=\!0$ be its fixed point.
The generalization to the case of other values of $L$ is straightforward, so, here I consider only case $L=0$. Let $S=T'(0)$. 
The schr\"oderfunction is solution $g$ of the Schr\"oder equation
\be
g(T(z))=S\, g(z)
\eL{schroederEq}
where $s$ is some constant. Usually, it is assumed that  $T(0)\!=\!0$, id est, zero is fixed point of the transfer function $T$.
One can search for solutions in the asymptotic form
\rf{schroederEq} 
\be
g(z)=\sum_{p=1}^{\infty} a_p \, z^p 
\eL{schroederexpa}
where coefficients $a$ are constants, id est, do not depend on $z$.
Usually, the series in the right hand side of
\rf{schroederexpa} 
diverges, but this representation can be used for approximation of $g$ at small values of 
$|z|$.
Then, any required precision can be achieved, applying the Schroeder equation
\rf{schroederEq} or its inverse
\be
g(z)= S \, g\Big(T^{-1}(z)\Big)
\eL{schroederEr}
looking, what is smaller, $|T(z)|$ or $|T^{-1}(z)|$.
This method is especially explicit for 
a real-holomorphic transfer function $T$, that monotonously grows along the real axis.
\vskip 6pt

In certain parti f the complex plane, the abelfunction $G$ can be expiressed through the Schroeder function $g$:
\be
G(z)=\log_S(g(z))
\eL{shroederabel}
Taking logarithm to base $S$ of the both sides of equation \rf{schroederEq}, I get
\be
G(T(z))=1+G(z)
\eL{analo}
this equation is just Abel equation for the same transfer function $T$.
\vskip 6pt

\begin{figure}[t]
\hskip 4pt
\sx{.333}{\begin{picture}(1252,1230) 
\put(50,20){\ing{olga6ma}} 
\put(14,1204){\sx{6}{$y$}}
\put(18,1116){\sx{5}{$5$}}
\put(18,1016){\sx{5}{$4$}}
\put(18, 916){\sx{5}{$3$}}
\put(18, 816){\sx{5}{$2$}}
\put(18, 716){\sx{5}{$1$}}
\put(18, 616){\sx{5}{$0$}}
\put(-22, 516){\sx{5}{$-1$}}
\put(-22, 416){\sx{5}{$-2$}}
\put(-22, 316){\sx{5}{$-3$}}
\put(-22, 216){\sx{5}{$-4$}}
\put(-22, 116){\sx{5}{$-5$}}
\put(-22, 016){\sx{5}{$-6$}}
\put( 18, -20){\sx{5}{$-6$}}
\put(118, -20){\sx{5}{$-5$}}
\put(218, -20){\sx{5}{$-4$}}
\put(318, -20){\sx{5}{$-3$}}
\put(418, -20){\sx{5}{$-2$}}
\put(518, -20){\sx{5}{$-1$}}
\put(649, -20){\sx{5}{$0$}}
\put(750, -20){\sx{5}{$1$}}
\put(850, -20){\sx{5}{$2$}}
\put(950, -20){\sx{5}{$3$}}
\put(1050, -20){\sx{5}{$4$}}
\put(1150, -20){\sx{5}{$5$}}
\put(1234, -20){\sx{6}{$x$}}
\put(170, 616){\sx{6}{\bf cut}}
\put(1060, 614){\sx{6}{$v\!=\!0$}}
\put(710, 1086){\sx{6}{\rot{75}$v\!=\!1$\ero}} 
\put(660, 222){\sx{6}{\rot{-77}$v\!=\!-1$\ero}} 
\put(86,790){\sx{6}{\rot{-29}$v\!=\!2$\ero}} 
\put(100, 438){\sx{6}{\rot{28}$v\!=\!-2$\ero}} 
\put(78, 1050){\sx{6}{\rot{37}$u\!=\!2$\ero}}
\put(460, 668){\sx{6}{\rot{38}$u\!=\!1$\ero}}
\end{picture}} 
\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Olga6map.jpg}
\vskip -4pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{Olga}(x\!+\!\mathrm i y)$ 
by \rf{olga}
\label{olgamap} }
\vskip 2pt
\end{figure}

\begin{figure}[t]
\hskip 4pt
\sx{.333}{\begin{picture}(1252,1230) 
\put(50,20){\ing{anka616ma}} 
\put(18,1210){\sx{5}{$y$}}
\put(22,1116){\sx{4.5}{$5$}}
\put(22,1016){\sx{4.5}{$4$}}
\put(22,916){\sx{4.5}{$3$}}
\put(22,816){\sx{4.5}{$2$}}
\put(22, 716){\sx{4.5}{$1$}}
\put(22, 616){\sx{4.5}{$0$}}
\put(-16, 516){\sx{4.5}{$-1$}}
\put(-16, 416){\sx{4.5}{$-2$}}
\put(-16, 316){\sx{4.5}{$-3$}}
\put(-16, 216){\sx{4.5}{$-4$}}
\put(-16, 116){\sx{4.5}{$-5$}}
\put(-16, 16){\sx{4.5}{$-6$}}
\put(18, -20){\sx{4.5}{$-6$}}
\put(118, -20){\sx{4.5}{$-5$}}
\put(218, -20){\sx{4.5}{$-4$}}
\put(318, -20){\sx{4.5}{$-3$}}
\put(418, -20){\sx{4.5}{$-2$}}
\put(518, -20){\sx{4.5}{$-1$}}
\put(650, -20){\sx{4.5}{$0$}}
\put(750, -20){\sx{4.5}{$1$}}
\put(850, -20){\sx{4.5}{$2$}}
\put(950, -20){\sx{4.5}{$3$}}
\put(1050, -20){\sx{4.5}{$4$}}
\put(1150, -20){\sx{4.5}{$5$}}
\put(1234, -20){\sx{5}{$x$}}
\put(186,1180){\sx{5}{\rot{4}$u\!=\!0$\ero}} 
\put(184,1002){\sx{5}{\rot{5}$v\!=\!0$\ero}} 
\put(182,816){\sx{5}{\rot{3}$u\!=\!0$\ero}} 
\put(180, 616){\sx{5}{$v\!=\!0$}} % \put(1200, 614){\sx{6}{$v\!=\!0$}}
\put(180, 422){\sx{5}{\rot{-6}$u\!=\!0$\ero}} 
\put(180, 232){\sx{5}{\rot{-6}$v\!=\!0$\ero}} 
\put(180, 56){\sx{5}{\rot{-6}$u\!=\!0$\ero}} 
\put(564, 640){\sx{5}{\rot{59}$v\!=\!0$\ero}}
\put(652, 630){\sx{5}{\rot{38}$u\!=\!0$\ero}}
\put(752, 610){\sx{5}{\rot{52}$u\!=\!1$\ero}}
\put(1066, 899){\sx{5}{\rot{2}$v\!=\!16$\ero}}
\put(1066, 858){\sx{5}{\rot{3}$u\!=\!-16$\ero}}
\put(1066, 776){\sx{5}{\rot{1}$u\!=\!-16$\ero}}
\put(1066, 726){\sx{5}{\rot{2}$u\!=\!16$\ero}}
\put(1066, 648){\sx{5}{\rot{-1}$v\!=\!16$\ero}}%
\put(1066, 590){\sx{5}{\rot{1}$v\!=\!-16$\ero}}%
\put(1066, 518){\sx{5}{\rot{-1}$u\!=\!16$\ero}}
\put(1066, 470){\sx{5}{\rot{0}$u\!=\!-16$\ero}}
\put(1060, 390){\sx{5}{\rot{-3}$v\!=\!-16$\ero}}
\put(1060, 338){\sx{5}{\rot{-1}$v\!=\!16$\ero}}
\put(1040, 256){\sx{5}{\rot{-4}$u\!=\!-16$\ero}}
\put(1040, 200){\sx{5}{\rot{-2}$u\!=\!16$\ero}}
\put(1030, 106){\sx{5}{\rot{-5}$v\!=\!16$\ero}}
\put(1030, 60){\sx{5}{\rot{-3}$v\!=\!-16$\ero}}
\end{picture}} 
\vskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Anka616map.jpg}
\vskip -6pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{Anka}(x\!+\!\mathrm i y)$ 
by \rf{anka}
\label{ankamap} }
\end{figure}

In the similar way one can express the analogy of superfunction, it can be called ``scaling function'' or even ``scalingfunction''. 
As examples of the Schroeder functions and the scaling functions, figure \ref{olgamap} and \ref{ankamap}
show complex maps of functions Olga and Anka. They are Schroeder function and the scaling function for the transfer function $T=\mathrm{Doya}$, considered in chapter 5 and in the previous section.
\vskip 6pt

The Schroeder function Olga can be expressed through function Tania:
\be
\mathrm{Olga}(z) = \mathrm{Tania}\Big(\ln(z)\Big)
\eL{olga}
The Scaling function Anka can be expressed through the ArcTania: 
\be
\mathrm{Anka}(z) = \exp\!\Big(\mathrm{ArcTania}(z)\Big)
\eL{anka}
For the transfer function Doya, the scaling factor $S\!=\!\mathrm e$; so, the natural logarithm and the natural exponent appear in equations \rf{olga} and \rf{anka}.
Definitions \rf{olga} and \rf{anka} imply, that $\mathrm{olga}\!=\!\mathrm{anka}^{-1}$.
\vskip 6pt

In this section, I use the example, when some dependences can be expressed in terms of elementary functions.
In particular, this refers to function ArcTania by equation \rf{ArcTaniaz}; in the most of the complex plane (except the negative part of the real axis), $\mathrm{ArcTania}(z)=z+\ln(z)-1$. Then, function Anka can be expressed as follows
\be
\mathrm{Anka}(z) = \frac{z}{\mathrm e} \exp(z)
\eL{ankae}
Functions Anka and Olga satisfy the scaling equation 
\be
\mathrm{Doya}(\mathrm{Anka}(z))=\mathrm{Anka}(\mathrm e\, z)
\ee
and the Schroeder equation
\be
\mathrm{Olga}(\mathrm{Doya}(z))=\mathrm e \,\mathrm{Olga}(z)
\ee
The Reader is invited to investigate the ranges of validity of equations
\be
\mathrm{Olga}(\mathrm{Anka}(z))=z  \\
\mathrm{Anka}(\mathrm{Olga}(z))=z
\ee
In wide range, that includes the positive part of the real axis, iterates of function $T\!=\!\mathrm{Doya}$ can be expressed through its scaling function 
$f\!=\!\mathrm{Anka}$ 
and Schr\"oder function 
$g=\mathrm{Olga}=\mathrm{Anka}^{-1}\!=\!\mathrm{ArcAnka}\,$:
\be
T^n=f(S^n\, g(z))
\eL{Tnfg}
for some appropriate constant $S$.
Readers are invited to check, that formula \rf{Tnfg} gives the same 
iterates, as the expression through the superfunction $F\!=\!\mathrm{Tania}$ and Abel function $G\!=\!\mathrm{ArcTania}$
\be
T^n=F(n + G(z))
\eL{TnFG}
While the real-holomorphic transfer function $T$ has real fixed point $L$ with the scaling factor $S$, and this fixed point is used 
to construct the superfunction $F$ and the scalingfunction $f$, formulas 
\rf{Tnfg} and \rf{TnFG} are equivalent. 
Several examples from this book, considered with superfunctions and abelfunctions, 
can be treated also with scalingfunctions and schro\"oderfunctions.
\vskip 6pt

Relation \rf{TnFG} is more general than \rf{Tnfg}. Here I announce few cases, when the Schroeder functions fail.
\vskip 1mm
If $T'(L)\!=\!1$, then the scaling factor $S$ becomes unity, and expression \rf{shroederabel} fails.
In particular, this is case of $T(z)\!=\!\exp(z/\mathrm e)$, case of $T(z)\!=\!\mathrm{zex}(z)\!=\!z\,\exp(z)$, and that for  $T(z)\!=\!\sin(z)$,
see Table \ref{TableOfSuperfunctions}.
\vskip 2pt

The transfer function $T$ may have no real fixed points, as it takes place for $T\!=\!\exp$.
\vskip 2pt

In addition, it may happen, that the transfer function $T$ has no fixed points at all, as it takes place for
$T(z)\!=\!\mathrm{tra}(z)\!=\!z\!+\!\exp(z)$.
\vskip 2pt

Iterates of these functions $T$ are straightforward with the superfunctions and abelfunctions.
These examples are mentioned in the Table \ref{TableOfSuperfunctions} and considered in the following chapters of this Book.
However, these cases are difficult to treat with the scalingfunctions and the schroederfunctions, if at al.
The Schroeder functions, if they can be applied, do not give any new in compare to use of the superfunctions.
For this reason, this Book is dedicated to superfunctions and abelfunctions, and not to schroederfunctions.
\vskip 6pt
I hope, the example above is sufficient to feel relation between Abelfunctions and Schroederfunctions.
On this point I stop speculations about the Schroeder functions, Schroeder equations, Scaling equations, and scaling functions,
and return to superfunctions. Superfunctions for the specific quadratic transfer function are considered in the next chapter.

\JP{



\JP{На этом я заканчиваю спекуляции на тему Шрёдера и возращаюсь к суперфункциям, абельфункциям и итерациям. В следующей главе рассмотрены регулярные итерации относительно простой квадратичной функции, а именно - логистического оператора, или логистического отображения.}%

\begin{wrapfigure}{r}{174px}
\begin{center}
\normalsize
\vskip 2pt
\sx{.67}{\begin{picture}(200,110)
\put(0,0){\ing{monoxid}}
\put(84,64){\sx{4}{O}}
\put(24,30){\sx{4}{H}}
\put(144,30){\sx{4}{H}}
\end{picture}}
\end{center}
\vskip -18pt
\caption{dihydrogen monoxid \label{mnx}\label{monoxid}}
\end{wrapfigure}

As continuation about superfunctions,
in the next chapter I shall consider the logistic map as the transfer function.
Term ``logistic map'' looks absurdly long,  but ``logistic map'' is used in the literature to name the function considered in the next chapter; and I cannot yet justify any more suitable name. Actually, it is just a special kind of the quadratic function.
One should not afraid of it more than of the ``dangerous chemical'' dihydrogena monoxide shown in figure \ref{monoxid}.
}%JP

\chapter{ ~ Logistic map}
\baselineskip 19pt
\begin{wrapfigure}{r}{60px}
\sx{.3}{\begin{picture}(110,120)
\put(0,0){\ing{Elutin}}
\end{picture}}
\vskip -4pt
\caption{\baselineskip 11pt
~ P.V.Elutin \label{elutinFig}}
\vskip -4pt
\end{wrapfigure}

Term ``Logistic map'' 
\footnote{\url{https://en.wikipedia.org/wiki/Logistic_map}}
may refer to the quadratic transfer function 
\be
T(z)=\mathrm{Elu}_s(z)=s\, z \, (1\!-\!z)
\eL{logisticop}
Usually, parameter $s$ is assumed to be a real number.
Term ``map'' is used also in the context of the ``complex map'', and this may cause confusions.
For this reason I give this function name $\mathrm{Elu}_s$. The thee characters of the name are taken from the last name of Pawel Elutin (see Fig. \ref{elutinFig}), my teacher of Quantum Mechanics. He asked me to 
construct the analytic extension 
of iterates of function $T$ by \rf{logisticop} in the private communication \cite{elutin}. In such a way, this chapter could be called also ``Elutin function''. Here I present some results of publication \cite{logistic} that appeared as the answer on the request by Elutin.
\vskip 2mm

Historically, iterates of another transfer function had been considered before the iterates of the Elutin function. I mean, iterates of factorial \cite{factorial}. The request to do the same of the logistic map appeared as result of the publication about the half iterate of factorial. In this Book, I do not follow the history; %(I am interested in math rather than in its history)
first, I describe the superfunction, abelfunction and iterates of the logistic map, or iterates of the Elutin function by \rf{logisticop}; these iterates seem to be simpler than those for the factorial. For various values of parameter $s$, iterates of $\mathrm{Elu}_s$ are shown in figure
\ref{figlogi1}. In order to plot this figure, I use the specific superfuction
and the corresponding abelfunction. I describe them in this chapter.
\vskip 2mm

The logistic sequence is solution $F$ of the logistic equation
\be
F(z\!+\!1)=\mathrm{Elu}_s(F(z))
\eL{LOGEQ}
Equation \rf{LOGEQ} is, actually, the transfer equation \rf{transfereq}, with the Elutin function (logistic map) $\mathrm{Elu}_s$ by \rf{logisticop} as the transfer function $T$.
In order to define the sequence, the initial value $F(0)$ should be specified.
\vskip 2mm

\JP{I do not think, that names ``logistic map'' and ``Logistic sequence'' are very convenient notations. Perhaps, I should suggest shorter and less confusive names for them. The colleagues are cordially invited to donate their name to call these functions
\footnote{Were I be less arrogant, I should accept suggestion by Henryk Trappman to call some functions with my names. However, I decline such an idea, because the number of functions I need to name greatly exceeds the number of my names, that, in first approximation, is about of 4: first name, last name, patterno (``othestvo'', отчество) and materno (матерно).}.
While I use the ``neutral'', generic  notations $T$ for the logistic map, letter $F$ to denote its superfunction, id est, any holomorphic solution of equation \rf{LOGEQ}, and $G$ to denote the corresponding abelfunction, $G=F^{-1}$.
\vskip 2mm
}%JP

In publications about the logistic equation \rf{LOGEQ}  with transfer function
\rf{logisticop}, 
the argument of function $F$ is assumed to be an integer number
\cite{L1992,L2000,L2003,L2009}. 
For integer argument of the solution $F$ of equation \rf{LOGEQ}, the solution
catches some properties of transition of the physical systems to chaos
\cite{L1,L2,L3}. 
Iterates of the transfer function $T$ by \rf{logisticop} 
appear as a rough description of the stochastic physical systems, as a simple, heuristic approach for the problems of hydro- and aero- dynamics, and also the transition regime of the stochastic lasers, in vicinity of the single-mode regime of generation.
Here I consider the holomorphic extension of the logistic sequence for not only integer, but complex values of the argument.
\vskip 2mm

\JP{My teacher of Quantum Mechanics, I mean Pawel Elutin, knew about my ambitions about superfunctions, and asked me to built-up the continual generalisation of the logistic sequence
\cite{elutin}, 
and this generalisation had been constructed and reported \cite{logistic}. The main results of that publication are presented in this chapter.}%JP

\JP{ Исторически, сперва была построена голоморфная суперэкспонента (тетрация) по основанию $\mathrm e$, потом - тетрация по основанию $\sqrt{2}$, потом - суперфакториал, и только потом - голоморфное расширение логистической последовательности. Однако логистический оператор, который является квадратичной функцией, во многих отношениях проще, чем факториал и экспонента. Кроме того, для $s=4$, голоморфное расширение логистической последовательности, то есть суперфункция для передаточной функции $T$ по формуле \rf{logisticop}, выражается элементарной функцией.
Я занимаюсь физикой и математикой, но не историей физики и математики. Поэтому, вместо того, чтобы излагать результаты в последовательности, в которой они были получены, я начинаю с простых примеров и потом перехожу к более сложным. }%JP

\section{Logistic sequence}
\baselineskip 19.2pt

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\newcommand \capOne{
Iterates of the Elutin function
\rf{logisticop}: $y\!=\! \mathrm{Elu}_s^{n}(x)$
for $s\!=\!3$, left; 
for $s\!=\!4$, center; 
for $s\!=\!5$, right; curves for 
$n\!=\!1$,
$n\!=\!0.8$,
$n\!=\!0.5$,
$n\!=\!0.2$ are drawn. 
\JP{
для $s\!=\!3$, слева; 
для $s\!=\!4$, по центру; 
для $s\!=\!5$, справа}
\label{Logi1a345T300}
}
\begin{figure}
\figOne 
\vskip -4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi1a345T300.png}
\vskip -6pt
\caption{\capOne~
\label{eluite}
\label{figlogi1} }  
\vskip 24pt
\end{figure}

Iterates of the Elutin function (logistic map)  $T\!=\!\mathrm{Elu}_s$ by \rf{logisticop}, 
as functions of real argument, are shown in figure \ref{figlogi1} 
for $s\!=\!3$ (left picture) $s\!=\!4$ (central picture) and $s\!=\!5$ (right picture); 
$y\!=\! \mathrm{Elu}_s^{\,n}(x)$ is plotted versus $x$ for $n\!=\!0.2$, $n\!=\!0.5$, $n\!=\!0.8$ and $n\!=\!1$.
All the graphics in figure  \ref{figlogi1} are plotted with the same formula
\be
\mathrm{Elu}_s^n(z)=F(n+G(z))
\eL{ite}
where $F$ is specific superfunction, solution of equation \rf{LOGEQ}, and  $G\!=\!F^{-1}$ 
is the inverse function (id est, abelfunction of $\mathrm{Elu}_s$).
Superfunction $F$ 
appears
as holomorphic extension of the logistic sequence \cite{logistic}.
Superfunction $F$ can be constructed with the regular iteration, described in the previous chapter.
This construction is described below.
\vskip 2mm

At the construction of a superfunction, the key question is about the fixed points of the transfer function.
For transfer function $T\!=\!\mathrm{Elu}$ by \rf{logisticop}, equation $\mathrm{Elu}_s(z)\!=\!z$ has two solutions,
$z\!=\!0$ and $z\!=\!1\!-\!1/s$. 
First of these solutions does not depend on $s$. This solution is used for construction and evaluation of the ``holomorphic extension of the logistic sequence'', id est, the specific  solution $F=\mathrm{Pav}_s$ of equation \ref{LOGEQ},
shown in figure \ref{logi56} below.
Then I can construct the inverse function
$G=F^{-1}$, which is the corresponding abelfunction. These $F$ and $G$ allow to plot figure  \ref{figlogi1} by formula
\rf{ite}. 
Construction of these functions
is described in the next section.

\section{Fixed point $L\!=\!0$}
\baselineskip 18.4pt

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\vskip  10pt

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\end{picture}}
\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:LogisticSecK2.jpg}
\vskip -10pt
\caption{\large $\mathrm{Pav}_s(x)$
by formulas \rf{as},\rf{regi}, \rf{logilim} for $s\!=\!3$, $s\!=\!3.4$, $s\!=\!3.8$, at the top picture, and for $s\!=\!3.9$, $s\!=\!4$, $s\!=\!4.1$, bottom.
\iL{logi56}}
\vskip 18pt
\end{figure}

For the 
Elutin transfer function by \rf{logisticop}, with the regular iteration at the fixed point $L\!=\!0$, I construct the superfunction $F=\mathrm{Pav}_s$. Graphics $y\!=\! \mathrm{Pav}_s(x)$ are shown in figures \ref{logi56} for different $s$. 
The complex maps are shown in figures \ref{logi2c3}, \ref{logi2c4}, \ref{logi2c5}.
Below I describe the construction and evaluation of this superfunction. 
\vskip 2mm

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\vskip -2pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi2c3T1000.png}
\vskip -4pt
\caption{ \large
$u\!+\!\mathrm i v \!=\! \mathrm{Pav}_3(x\!+\! \mathrm i y)$ by \rf{logilim}
\iL{logi2c3}
}
\vskip 2pt
\end{figure}

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\vskip -2pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi2c4T1000.png} 
\vskip -4pt
\caption{ \large
$u\!+\!\mathrm i v \!=\! \mathrm{Pav}_4(x\!+\! \mathrm i y)$ by \rf{logilim}
\iL{logi2c4}
}
\vskip 2pt
\end{figure}

\begin{figure}[t]
\normalsize
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\hskip 36pt
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\vskip -4pt
\caption{ \large
$u\!+\!\mathrm i v \!=\! \mathrm{Pav}_5(x\!+\! \mathrm i y)$ by \rf{logilim}
\iL{logi2c5}
}
\vskip 2pt
\end{figure}
\vskip 2mm

I need some name to denote superfunction for the transfer function $\mathrm{Elu}_z$; For this superfunction, I suggest name ``Pav''. In this name, I use first 2.5 characters of the first name Pawel of my teacher (who had asked me to construct this function; half of letter w appears as v). First three characters of his last name Elutin are already used below to denote the quadratic function
\rf{logisticop}, known also as ``logistic map''. My excuse for defining of the new name is the following: the ``logistic map'', as it is called in the literature, is not actually map in the common sense of this word;
it is holomorphic function; and the similar note refers to the so-called ``logistic sequence''. So, I use term ``Elutin function'' instead of ``logistic map''; in the context of this book, it is holomorhic function, just some polymomial.
\vskip 2mm

In order to define function $\mathrm{Pav}_s$, first, I construct its asymptotic.
For the fixed point $L\!=\!0$,  in the expansion \rf{Lea2general}, increment $k\!=\!\log s$, and the expansion parameter  $\varepsilon\!=\!s^z$; 
This gives the primary expansion of the superfunction
\be
\tilde F(z)= \sum_{n=1}^{N-1} a_{n} s^{nz} + \mathcal{O}(s^{Nz})
\eL{as}
For the representation \rf{Lea2general}, I set also %также
 $a_1\!=\!1$.
Variation of this parameter causes only the displacement of the argument of the resulting superfunction; this does not affect the iterates of the transfer function $\mathrm{Elu}_s$.
Expressions  \rf{a2.39} determine the coefficients $a$. In particular,
\be
a_{2}&=& \frac{1}{1\!-\!s}  ~~~~~~~~~ ~~~~~~~~ ~~~~~~~ ~~~~~ \iL{logia2}\\
a_{3}&=& \frac{2}{(1\!-\!s)(1\!-\!s^2)} ~~~~~~~~~~ ~~~~~~~\iL{logia3}\\
a_{4}&=& \frac{5+s}{(1\!-\!s)(1\!-\!s^2)(1-s^3)} ~~~~~~\iL{logia4}
\ee
The primary representation  $\tilde F(z)$  by \rf{as} allows the accurate (precise) evaluation of $F(z)$ at large negative values of $\Re(z)$.
Then, the continual extension of the logistic sequence,
shown in figure \ref{logi56}, appears as limit
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F(z)= \mathrm{Pav}_s(z) 
=
\lim_{n\rightarrow \infty} \mathrm{Elu}_s^{~n}\Big(\tilde F(z\!-\!n)\Big) ~ ~
\iL{logilim}
\ee

For real 
argument, graphics of superfunction  $F\!=\!\mathrm{Pav}_s$ are shown in figure \ref{logi56} for $s\!=\!3$, $s\!=\!3.4$, $s\!=\!3.8$ at the top picture and for $s\!=\!3.9$, $s\!=\!4$, $s\!=\!4.1$ at the bottom picture.
For $s\!=\!3$, $s\!=\!4$ and $s\!=\!5$, the complex maps of function $F\!=\!\mathrm{Pav}_s$ are shown in figures 
\ref{logi2c3},
\ref{logi2c4},
\ref{logi2c5} announced above.


While  $s\!<\!3.5$ (id est, does not exceed the ``Pomequ-Mannevill constants'' \cite{logistic,pomeau,gac}  ), 
the logistic sequence $F$ (as for integer values of the argument, as for the real values) shows pretty boring and regular oscillations.
At larger values of $s$, at the argument grows, the oscillations become dense.
Observation of $F(n)\!=\!\mathrm{Pav}_s(n)$ at the integer values of $n$ make an impression of quasi-random sequence. 
While  $s\!\le\!4$, at real $x$, function $F(x)$ oscillates within segment [0,1], and only at $s\!=\!4$ it touches the borders of this segment.
At $s\!>\!4$, the function has ``gaps''; they become deeper and deeper at the increase of the argument $x$ or parameter $s$.
\vskip 2mm

At $s\!=\!4$, the holomorphic extensiion $F_s\!=\!\mathrm{Pav}_s$ of the logistic sequence can be represented as the elementary function,
\be
\mathrm{Pav}_4(z)=(1-\cos(2^z))/2
\eL{logicos}
The curve, corresponding to $s\!=\!4$ at figure \ref{logi56}, could be plotted even without the regular iteration; the same applies to map at figure \ref{logi2c4}.
\vskip 2mm

At consideration of the holomorphic extension of the logistic sequence in the complex plane, its behaviour is regular.
However, at large $s$, the oscillations become more and more dense with the increase of the argument.
\vskip 2mm

Superfunction $F_s$, built with regular iteration at the fixed point $L\!=\!0$, is entire and periodic.
The period is pure imaginary (for real $s$):
\be
P= 2 \pi \rm i / \ln(s)
\eL{logiperiod}
This periodicity is seen in figures \ref{logi2c3}, \ref{logi2c4}, \ref{logi2c5}: 
The isolines are reproduced at the corresponding translations along the imaginary axis. A little bit more than one period fit the height of the map in figures
\ref{logi2c3}, \ref{logi2c4}, \ref{logi2c5}.
\vskip 2mm

With holomorphic extension of the logistic sequence, id est, superfunction $\mathrm{Pav}_s$, one can construct the non-integer iterates of the logistic map, id est, iterate the Elutin function $\mathrm{Elu}_s$ by  \rf{logisticop}; in particular, the half iterate of this transfer function can be constructed. However, for this, the inverse function is also required, I mean, the abelfunction
$G_s\!=\!\mathrm{ArcPav}_s\!=\!\mathrm{Pav}_s^{-1}$.
\vskip 2mm

The complex maps of the abelfunction $\mathrm{ArcPav}_s$ are shown in figures
\ref{logi2d3}, \ref{logi2d4}, \ref{logi2d5} for
$s\!=\!3$,
$s\!=\!4$ and
$s\!=\!3$.
This abelfunction is described in the next section.

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\axes \end{picture}}
\vskip 2pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi2d3t1500.jpg}
\caption{\large 
$u\!+\!\mathrm i v = \mathrm{ArcPav}_3(x\!+\! \mathrm i y)$ %для $s\!=\!3$ 
\iL{logi2d3}}
\vskip 18pt
\end{figure}

\section{Abelfunction for the Elutin function}
\baselineskip 19.6pt
\begin{figure}[t]
\normalsize
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\axes \end{picture}}
\vskip 2pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi2d4t1500.jpg}
\caption{\large
$u\!+\!\mathrm i v = \mathrm{ArcPav}_4(x\!+\! \mathrm i y)$.
\iL{logi2d4}
}
\vskip 14pt
\end{figure}
\begin{figure}[t]
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\end{picture}} 
\vskip -4pt
%\url{https://mizugadro.mydns.jp/t/index.php/File:Logi2d5t1500.jpg}
\caption{\large %по формулам \rf{GC},\rf{Git} для  $s\!=\!5$;  
$u\!+\!\mathrm i v =  \mathrm{ArcPav}_5(x\!+\! \mathrm i y)$.
\iL{logi2d5}
}
\vskip -2pt
\end{figure}
This section describes the 
Abel function of the Elutin function $\mathrm{Elu}_s$ by \rf{logisticop}. I call this function $\mathrm{ArcPav}_s$.
It is inverse function of function $\mathrm{Pav}_s$ by \rf{logilim}, described in the previous section.
\vskip 8pt

Function $G\!=\!G_s\!=\!\mathrm{ArvPav}_s\!=\!\mathrm{Pav}_s^{-1}$ satisfies the Abel equation
\be
G_s(\mathrm{Elu}_s(z))=G_s(z)+1
\eL{GT}
This equation is the same as \rf{abeleq}, the only additional subscript $_s$ is gaffed to indicate the parameter in the transfer function $T=\mathrm{Pav}_s$.
Complex maps of function $G_s$ are shown in figures 
\ref{logi2d3}, \ref{logi2d4}, \ref{logi2d5} for various values of $s$.
This section describes evaluation of this function.
\vskip 8pt

The asymptotic expansion $\tilde G$ for abelfunction $G$ can be found with inversion of expansion \rf{as} for superfunction $F$; it has the following form:
\be
\tilde G(z)&=&\log_s \left( \sum_{n=1}^{N-1} C_n z^n +O(z^{N})\right)  \iL{GC}
\ee
In the software Mathematica, there is special procedure  {\bf InverseSeries} for such inversions.
\vskip 8pt

Coefficients $C$ in equation \rf{GC} depend on parameter $s$, but this dependence is not indicated explicitly, in order to keep the expression compact.
The same coefficients can be found also substituting the expansion
\rf{GC} 
into the Abel equation
\rf{GT} with transfer function $\mathrm{Elu}_s$
and equalizing the coefficients at equal powers of $z$ in the left and in the right hand sides.
The resulting coefficients $C$ appear as follows 

\be
C_1&=&1 \\
C_2&=& \frac{1}{s-1}\\
C_3&=& \frac{ 3s}{ (s-1)(s^2-1)}\\
C_4&=& \frac{ (s^2\!-\!5)s}{ (s\!-\!1)(s^2\!-\!1)(s^3\!-\!1)}\\
... \nonumber
\ee

~ \baselineskip 18pt

The truncated series of expansion \rf{GC} gives way to evaluate abefunction $G$ at small values of the argument.
At large values, the representation can be extended with
\be
G(z)\approx \tilde G(\mathrm{Elu}_s^{-n}(z))+n
\eL{Git}
for some large enough integer $n$.
The negative integer iterates of the transfer function can be evaluated, using representation of the inverse of the transfer function:
\be
\mathrm{ArcElu}_s^{-1}(z)=\mathrm{ArcElu}_s(z)=\frac{1}{2} - \sqrt{\frac{1}{4}-\frac{z}{s}}
\label{logiTm1}
\ee
Symbol ``$\approx$'' in expression \rf{Git} should not make an impression, that only the approximation of the Abel function is constructed.
The exact Abel function can be expressed through the limit
\be\!\!\!\!\!\!\!\!
\mathrm{ArcPav}_s(z)=G(z)=\lim_{n\rightarrow \infty} \Big( \tilde G(\mathrm{Elu}_s^{-n}(z))+n \Big) ~ ~ ~
\label{AbelLogistic}
\ee
where symbol
$\tilde G_s$ 
denotes the truncated series in the expansion
\rf{GC}.
This representation is used in generators of figures
\ref{logi2d3}, 
\ref{logi2d4}, 
\ref{logi2d5}~. 
The same representation is used to plot iterates of the transfer function in figure
\ref{figlogi1}.
at
$s\!=\!3$,
$s\!=\!4$ and
$s\!=\!5$. 
\vskip 2pt

Function $\mathrm{ArcElu}_s(z)$ by \rf{logiTm1} has the branch point $z\!=\!s/4$. 
This determines also the branch of the abelfunction $\mathrm{ArcPav}_s$ and cuts in the right hand sides of maps in figures
\ref{logi2d3}, 
\ref{logi2d4}, 
\ref{logi2d5}~. 

The non-integer iterates of the transfer function, shown in figure \ref{Logi1a345T300}, have the similar branching.
The half iterate is special case of the non-integer iterate; this case shown in figures 
\ref{logi2b3},
\ref{logi2b4},
\ref{logi2b5} and described in the next section.

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\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi2b3t1000.jpg}
\vskip -2pt
\caption{\large %Карта функции %$h\!=\! \mathrm{LogisticOperator}_3^{0.5)$; 
\large $u\!+\!\mathrm i v = \mathrm{Elu}_3^{1/2}(x\!+\! \mathrm i y)$.
\iL{logi2b3}}
\vskip 4pt
\end{figure}

\section{Halfiterate of logistic operator %Половинная итерация %логистического оператора
}
\baselineskip 20.1pt

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\vskip 1pt
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\vskip -6pt
\caption{ \large
$u\!+\!\mathrm i v = \mathrm{Elu}_4^{0.5}(x\!+\! \mathrm i y)$
\iL{logi2b4}
}
\vskip 14pt
\end{figure}

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\vskip -4pt
\caption{ \large
$u\!+\!\mathrm i v = \mathrm{Elu}_5^{0.5}(x\!+\! \mathrm i y)$
\iL{logi2b5}
}
\vskip 14pt
\end{figure}

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\caption{\large $u\!+\! \mathrm i v =$ $-1.8(-(x\!+\!\mathrm i y))^{\sqrt{2}}$ \label{wrap37}} \vskip 12pt
\end{figure}
With superfunction $F_s$ and abelfunction $G_s=F_s^{-1}$, 
iterates of the transfer function $T_s$ are expressed with formula \rf{ite}. This representation is used to generate figure
 \ref{figlogi1}. 
In particular, $T_s^{0.5}$ by \rf{ite} is solution of the problem, formulated by P.Elutin:
The function $h_s$ is built as
\be \!\!\!\!
h_s\!=\! \mathrm{Elu}_s^{0.5}
~ ~ ~ ~ ~
\eL{logih}
such that its second iterate gives the logistic operator \rf{logisticop}; 
id est, for some range of values of $z$, the relation below holds:
\be \!\!\!\!\!
h^2(z)=h(h(z))=T(z)=\mathrm{Elu}_s(z)=s\,z\,(1\!-\!z) ~
\label{h2}
\ee
Complex maps of function $h_s$ are shown in figures
\ref{logi2b3},
\ref{logi2b4} and
\ref{logi2b5} 
for various values of parameter $s$.
These maps look similar, because, at large values of the argument $z$, the approximate relation takes place: 
\be
\mathrm{Elu}_s(z)\approx  \tau(z)=-s (-z)^2
\eL{lopbig}
Iterates of function $\tau$ by  \rf{lopbig}
can be calculated in analogy with iterates of the transfer function by 
\rf{Pow}. 
\vskip 2mm

The readers are invited to plot the superfunction and the abelfunction for the transfer function $\tau$.
This can be done in analogy with the iterates of the power function, presented in Table \ref{TableOfSuperfunctions}.
In particular, the half iterate can be written as follows:
\be
\tau^{1/2}(z)=-\alpha (-z)^{\sqrt{2}}
\eL{logita}
where  $\alpha=s^{1/(1+\sqrt{2})}$ 
is constant, and this constant depends on parameter $s$ slowly.

Figure \ref{wrap37} shows the map of function $\tau^{1/2}$ by formula \ref{logita} at $\alpha=1.8$; 
it should be compared to maps 
\ref{logi2b3},
\ref{logi2b4} and
\ref{logi2b5}~, 
that correspond to various values of $s$. 
At large values $x^2\!+\!y^2$, the only quadratic term in the expansion of the transfer function is important,
and all the four maps at figures  \ref{wrap37},
\ref{logi2b3},
\ref{logi2b4} and
\ref{logi2b5}
look similar.
In order to make the difference seen, these maps occupy all the width of the page available.
\vskip 2mm

When the non-integer iterate of some function is constructed, it has sense to check the results of iteration of this iterate.
For the transfer function 
$\mathrm{Elu}_s$ and its half iterate $h\!=\!\mathrm{Elu}_s^{0.5}$,
such a check, test is shown in figure 
\ref{logiha300}. 
This figure shows the map pf function $h^2$ for 
$s\!=\!3$, $s\!=\!4$ and  $s\!=\!5$, 
id est, for the same values of parameter, that are used in figure \ref {Logi1a345T300}.

\vskip 2mm
In the left hand side of maps in figure  \ref{logiha300},
the second iterate 
of function $h$ coincides with the logistic mapping.
The scratched lines show the limit of applicability of relation
\rf{h2}.
\vskip 2mm

Relation $h(h(z))\!=\!T(z)$ is valid at least for $\Re(z)\!<\!1/2~$. 
In such a way, the holomorphic extension of the logistic sequence, and also the corresponding non-integer iterates of the 
logistic operator correspond to the intuitive expectations about these functions.

\vskip 2mm
There is nothing specific in the half iterate of a function, while it is expressed through the superfunction and the abelfunction. It is just non-integer iterate, while number $n$ of iterate is 1/2.
On the other side, in the literature, the iteration half is often considered as something magic; perhaps, because for iterate half of some transfer function, the verification is especially simple; just iterate the halfiterate twice and see the region, where the result coincide with the original transfer function. Figure \ref{logiha300} appears as an example of such a verification.

\vskip 2mm
Interest to iterate half, since the half iterate of factorial ($\sqrt{\,!\,}$) \cite{kandi,sado} appears as a tradition.
Namely iterate half is mentioned in the titles of publications \cite{kneser,paulsen2016}. 
So, I follow this tradition and analyse range of validity of representation of a function through its iterate half.



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\end{picture}}
\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logiha300.jpg}
\vskip -1pt
\caption{\large
$u\!+\!\mathrm i v\!=\!h(h(x\!+\!\mathrm i y))$
by
\rf{logih}
for
$s\!=\!3$, 
$s\!=\!4$ and 
$s\!=\!5$ 
 \label{logiha300}}
\end{figure}

\section{Another fixed point, $L=1-1/s$}
\baselineskip 17.2pt

Holomorphic extension $F_s$  of the logistic sequence by formulas \rf{as},\rf{regi}
is regular and periodic; the period $P_a$ by
\rf{logiperiod} 
is pure imaginary and slowly (as logarithm) depends on parameter $s$. 
This solution is not unique.
In analogy with the solution, that approaches the fixed point $L\!=\!0$,
one may construct other solutions, that approach the fixed point $L\!=\!1\!-\!1/s$ at minus infinity.
Complex map of one of such solutions is shown in figure
\ref{inaya}; this superfunction is described in the is section. 
\vskip 2mm
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\end{picture}}
\vskip -4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Logi5ab400.jpg}
\vskip -4pt
\caption{
Superfunction $\mathrm{Paw}_4(x)$ by  \rf{L11s},\rf{regi}: explicit plot $y\!=\!\mathrm{Paw}_4(x)$, top picture, and complex map
$u\!+\!\mathrm i v=\mathrm{Paw}_4(x\!+\!\mathrm i y)$, bottom picture, by \rf{inoelim}
\label{inaya}
\iL{logi5ab4}
}
\vskip 8pt
\end{figure}

The logistic operator, id est, the transfer function $T_s$ by \rf{logisticop} has two fixed points, $L\!=\!0$ and  $L=1-1/s$. Consider the last of these fixed points. By the general way of regular iterations, I construct the superfunction $f$, that approaches $1\!-\!1/s$ at minus infinity, as follows:
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
\tilde f(z)\!&\!=\!&\frac{s-1}s
+\sum_{n=1}^{N-1} d_n \, \Big( (s\!-\!2)^z \cos(\pi z+\varphi) \Big)^{\!n} \iL{logina}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
f(z)\!&\!=\!&\!\mathrm{Paw}_s(z) =\tilde F(z) + O\!\Big( (s\!-\!2)^z \cos(\pi z+\varphi) \Big)^{\!N}
~ ~\eL{L11s} 
where $d$ are real parameters and $\varphi$ is real constant.
Substitution of this expansion into equation \rf{transfereq} gives the chain of equations for coefficients $d$. As in the previous cases, 
I set
$d_{1}=1$; 
then
\be
d_2&=&\frac{-s}{(s-1)(s-2)}  \nonumber \\
d_3&=&\frac{-s^2}{(s-1)(s-2)(s-3)}   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \iL{logid2}\\
d_4&=&\frac{-(s-7)^3\, s^3}{(s-2)(s-3)(s^3-8s^2+22s-21)} 
\nonumber
\ee
\vskip 2mm
The truncated series gives the accurate approximation of function $F(z)$, while the effective parameter of expansion, id est 
$(s\!-\!2)^z \cos(\pi z +\varphi)$ is small.
Truncation with 4 coefficients in  \rf{L11s} 
provides of order of 10 significant figures while
\be
\pi |\Im(z)| \!+\! \ln(s\!-\!2) \Re(z) <4
\ee
The range of approximation can be extended with iterations \rf{regi}; this can be used both for definition and algorithm of precise evaluation of 
the superfunction: 
\be
f(z)=\mathrm{Paw}_s(z)=\lim_{n\rightarrow \infty} \mathrm{Elu}_s^{\,n}(\tilde f(z\!-\!n))
\eL{inoelim}
I denote this function with $\mathrm{Paw}_s$, in order to distinguish it from $\mathrm{Pav}_s$ in the previous sections;
I hope, use of generic name $F$ for different superfunctions will not cause confusions. 
Representation \rf{inoelim} is used to generate figure \ref{logi5ab4}.
\vskip 2mm

Superfunction $\mathrm{Paw}_s$ by \rf{inoelim}  is asymptotically–periodic; the asymptotic period
\be
P=\frac{2\pi}{\ln(s\!-\!2)\, \mathrm i+\pi}
\eL{logiquasiP}
in the upped half plane and $P^*$ in the lower half plane.
\vskip 2mm

In contrast with superfunction $\mathrm{Pav}_s$,
that is built up at the fixed point zero, for superfunction $\mathrm{Paw}_s$,
the choice of the inverse function is not straightforward. One needs to choose, which of the oscillations should be used to return the value of function. %; and there are many options to set the cutlines. 
For this reason I do not provide the corresponding abelfunction here. The Reader is invited to construct it according to own preferences.
\vskip 2mm

Following the 6th of the TORI axioms, about the simplicity, I consider the superfunction by  
\rf{as},\rf{logilim} as ``principal'', because it seems to me simpler than 
that by \rf{logina}. 
For the logistic operator as transfer function, the regular iteration provides the regular superfunction; however, superfunctions, constructed at different fixed points, may show pretty different behaviour. 



\chapter{~ Factorial%and siperfactorial
} 
\baselineskip 20pt
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\caption{\large $y\!=\!\mathrm{Factorial}(x)$
\label{facsmall}
\label{facsma}
}
\vskip 4pt
\end{wrapfigure}

Factorial is holomorphic solution of equation
\be \!\!\!\!\!\!\!
\mathrm{Factorial}(z\!+\!1)=z\, \mathrm{Factorial}(z)
\ee
Factorial can be expressed with %integral
\be
\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{Factorial}(z)\!=\!
\int_0^\infty t^{z} e^{-t}\, \mathrm{d}t
\label{facdef}
\ee
I use this notation instead of that with with exclamation, $\mathrm{Factorial}(z)\!=\!z!$, in order to simplify indication of the number of iterate in the upper superscript.
\vskip 2mm

Graphic $y\!=\!\mathrm{Factorial}(x)$ versus $x$ is shown in figure \ref{facsmall}.
At $x\!>\!2$, factorial shows fast monotonous growth. At $0\!<\! n\! <\!1$, the iterates
$\mathrm{Factorial}^n$ should show the similar, but slower growth.
These iterates are topic of this chapter. 
\vskip 2mm

This chapter 
describes the superfunction of factorial, denoted below as SuFac, and the Abel function, denoted as AuFac.
Then, the iterates of Factorial appear as
\be
\mathrm{Factorial}^n(z)=\mathrm{SuFac}\big(n+\mathrm{AuFac}(z)\big)
\eL{faciterge}
Below, I construct functions SuFac and AuFac and describe their properties.
In this chapter, I retell the basic concept of publication in the Moscow University Physics Bulletin, 2010 \cite{factorial}.
\vskip 42mm

\section{Physics department}
Iterates of factorial and its superfunction and abelfunctions had been reported in 2010  \cite{factorial},
before iterates of the logistic map (Elutin function), described in the precious chapter. In this section I 
explain, why I consider this function as important.
\vskip 2mm

Past century, during the USSR, my teacher of Quantum Mechanics had asked students to give the physical sense to the operator
``square root of factorial''. That sign was well known in the USSR as symbol of the Physics Department of the Moscow State University
shown in figure \ref{SqrtFactorialLogo}, it is mentioned in the Introduction.
In analogy with other operators of Quantum Mechanics, 
``square root of factorial'' should be some function $h$ such that its second iteration gives factorial,  $h^2\!=\!\mathrm{Factorial}$, id est,
\be
h(h(z))\!=\!z!=\mathrm{Factorial}(z)
\eL{hhzfac}
One could guess, that this function should be real-holomorphic, growing faster than any polynomial but slower than any exponential.
That time it was difficult, to evaluate such a function $h$: the formalism of superfunctions had not yet been developed.
\vskip 2mm

In principle, the Schroeder function and the scaling function could be used to construct and evaluate the solution $h$ of equation \rf{hhzfac}, instead of superfunctions. 
That time, for students, dealing with Quantum Mechanics, it was difficult to guess, that for the square root of factorial, 
the Schroeder (Schr\"oder) equation should be used instead of the widely known Schroedinger (Schr\"odinger) one
\footnote{Iterates of factorial and $\sqrt{\,!\,}\,$\rule{0pt}{12pt} can be evaluated also with the scaling function $f$ and the Schroeder function $g$
by \rf{Tnfg}}. In addition, that time, no algorithms for evaluation of the scaling function and the Schroeder function were available.

\vskip 2mm

After the successful evalation of tetration to base 
$\sqrt{2}$, 
reported in 2009 at the journal Mathematics of Computation \cite{sqrt2}, 
the problem with iterates of factorial (and, in particular, of the half iterate) happen to be pretty solvable. The solution is published in the
Moscow University Physics Bulletin  \cite{factorial} and described below. 
\vskip 8pt

I feel, first I should remind the properties of factorial.
This is matter of the next section.


\section{Factorial and its fixed points}
\baselineskip 18.6pt

For real argument, the explicit plot of factorial is shown in figure \ref{facsma}. I extend a little bit that plot in figure
\ref{figfac}, and I add some other related functions. The complex map of factorial is shown in figure \ref{facmap}. 
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\vskip -16mm
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:FactoReal.jpg}
\vskip -4pt
\caption{\large 
Factorial and related functions for real argument
\iL{figfac}}
\vskip -200pt
\end{figure}

\rule{0pt}{200pt}

\baselineskip 18.8pt
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\vskip -2pt
\caption{\large Complex map: 
$u\!+\! \mathrm i v=\mathrm{Factorial}(x\!+\!\mathrm i y)$ \iL{facmap}
}
\end{figure}

\JP{
Factorial $T$ is solution of equation
\be
T(z\!+\!1)=z T(z)
\ee
holomorphic in the whole complex plane except the negative integer values of the argument.
Factorial of complex argument is built-in function in some programming languages. The solution can be expressed in the integral form
through the Gamma function,
\be
\Gamma(z)=\int_0^\infty t^{z-1} e^{-t}\, \mathrm{d}t
\ee
I would not say, that this form is somehow easier than 
\be
\mathrm{Factorial}(z)=\int_0^\infty t^{z} e^{-t}\, \mathrm{d}t
\eL{facdef}
and therefore I consider \rf{facdef} as definition of factorial.
\vskip 2mm
}%JP
For construction of superfunction of any transfer function, there is important question about its fixed points.
The fixed points of factorial are solutions $L$ of equation
$\mathrm{Factorial}(L)\!=\!L\,$.
For real values of argument, the explicit plot of factorial is shown in figure  \ref{figfac} with thick curve, $y\!=\!\mathrm{Factorial}(x)$.
The fixed points correspond to intersections of this curve with line $y\!=\!x$, also shown in the figure.
For comparison, the thin curves show functions $\mathrm{Factorial}^{-1}$ and $z \!\mapsto\! \mathrm{Factorial}(z)^{-1}$.
These curves are added in order to remind, that 
$ \mathrm{Factorial}^n(z)$,
$ \mathrm{Factorial}(z^n)$ and
$ \mathrm{Factorial}(z)^n$ have pretty different meanings.
Complex maps of factorial and ArcFactorial are shown in figures 
\ref{facmap} и \ref{arcfacmap}. 
\vskip 6pt

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\vskip -24pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:AfacmapT800.png}
\vskip -8pt
\caption{
$u\!+\! \mathrm i v=\mathrm{ArcFactorial}(x\!+\!\mathrm i y)$ \iL{arcfacmap}
}
\end{figure}
\vskip 2mm


I had found no C++ implementations of factorial and arcfactorial for complex arguments in the literature.
Mathematica software allows the evaluation, but does it a little bit slowly; the Maple software happened to be not better \cite{tea}, see also the appendix, chapter 22, section 2.
\vskip 6pt
 
For evaluation of superfunction and the abelfunction, the transfer function and/or its inverse should be evaluated many times,
and the efficient (quick and precise) implementation is important.
For this reason, the original ``complex double'' procedures for factorial and arcfactorial are suggested and loaded as\\
\url{https://mizugadro.mydns.jp/t/index.php/Fac.cin} ~ and \\
\url{https://mizugadro.mydns.jp/t/index.php/Afacc.cin}\\
These implementations are used to plot figures of this Chapter.
\vskip 6pt

In figure \ref{figfac}, I show also line $y\!=\!x$ and graphics
$y\!=\!\mathrm{Factorial}^{-1}(x)$ and $y\!=\!\mathrm{Factorial}(x)^{-1}$.
Some extremal points of factorial are shown; the local minimum at point 
$x\!=\!\nu_0$
and local maximum at point 
$x\!=\!\nu_1$; 
Values of factorial in these points are denoted as $\mu_0$ и $\mu_1$.
Function $y\!=\!\mathrm{Factorial}^{-1}(x)$ has the branch point  $x\!=\!\mu_0$, and
$\nu_0$ is its value 
at this point.
\JP{
В широкой области значений аргумента, комбинация факториала и аркфакториала эквивалентна тождественной функции, значение которой равно её аргументу. Эта функция показана на рисунке \ref{figfac} тонкой линией. Эта линия проведена на рисунке  \ref{figfac} не для того, чтобы напомнить читателю, что такое тождественная функция
\footnote{Предполагается, что это каждый младенец знает. У нас почему-то никто не знает, от чего умер Пушкин и чем фудоки отличаются от нибелунгов, а как очищают политуру \cite{erofeev} и что такое тождественная функция, и нас все знают.}, а чтобы показать, выделить вещественные стационарные точки факториала. Поэтому на рисунке \ref{figfac} прямая $y\!=\!x$ продолжена влево, где соотношение $\mathrm{Factorial}(\mathrm{ArcFactorial}(x))\!=\!x$ вызывает сомнения.
}
\vskip 6pt

\JP{
The frame of figure  \ref{figfac} covers 4 fixed points: $L\!=\!2$, $L\!=\!1$,  $L\!\approx\!-3.15$ and   $L\!\approx\!-3.95$; 
other real fixed points are out of range of  the figure.
In principle, each of these points can be used for the regular iteration.
In this Book, I choose only one fixed point,
$L\!=\!2$.
Namely this fixed point provides the real-holomorphic superfunction, that infinitely grows at plus infinity.
This superfunction is considered in the next section.
}%JP


\section{Regular iteration for factorial}
\baselineskip  17pt

The asymptotic expansion of superffactorial can be written as follows:
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F(z)&=&L+\exp(kx)+a_2 \exp(2kz)+a_3 \exp(3kz)+.. \nonumber \\
\!\!\!\!\!\!\!\!\!       &=&L+\varepsilon+a_2 \varepsilon^2+a_3 \varepsilon^3+ ..
\label{superfacass}
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \iL{sufass}
\ee
where $\varepsilon=\exp(kz)$, for some constant increment $k$ and constant coefficients $a$
According to the general formulas \rf{k38}, \rf{a2.39}, \rf{a2.40},
substitution of this expansion into the transfer equation
\be
\mathrm{Factorial}\big(F(z)\big)=F(z\!+\!1)
\eL{Tfac}
gives the value of the increment. For Factorial's fixed point $L\!=\!2$, I get
\be\!\!\!\!\!\!\!\!\!
k=
\ln\!\big(3\!+\!2\!~\fac'(0)\big)=\ln(3\!-\!2\,\gamma) 
  \approx 0.61278745233070836
~\label{fack}
\ee
where
$\gamma\!=\!-\Gamma'(1)\!
\approx\! 0.5772156649%0153286060651209008240243
$
is Euer constant. I set
$a_0\!=\!2$ and  $a_1\!=\!1$;
then the partial sum in 
expansion 
\rf{superfacass} 
is easy to program. I get
\be a_2&\!=\!& \frac{\pi^2+6\gamma^{2}-18\gamma+6}{12(3-5\gamma+2\gamma^{2})} \approx 0.798731835172434541585621%072345730147
\\
a_{3}&\!=\!&\! \Big( 
-36-39\pi^2
-738\gamma^2+
324\gamma+
99\pi^2\gamma-60\pi^2\gamma^2-\pi^4+24\gamma^5
\nonumber\\
&&+ 594\gamma^3-120\zeta(3)\gamma+48\zeta(3)\gamma^2+12\gamma^3\pi^2+72\zeta(3)-204\gamma^4 \Big) /  \nonumber\\
&& \Big(144(-18+69\gamma-104\gamma^2+77\gamma^3-28\gamma^4+4\gamma^5) \Big) 
\nonumber\\
&\approx & 0.5778809754764832358038%07592348110748
\label{u2u3zeta}
\ee
In equation \rf{u2u3zeta}, the Riemann zeta-function appears,
$\ds \zeta(z) \!=\! \sum_{n=1}^\infty \frac{1}{n^z} \,
$. %\eL{zeta}
We need $\zeta(3)\!\approx\! 1.202056903~$; 
values of function $\zeta$ at other arguments are not used for evaluation of coefficients $a$.
Approximate values of coefficients in expansion \rf{sufass} are shown in  table
\ref{CoefficientsAUFac}.
\begin{table}[t] %\label{table2} does not work
\begin{center}

\caption{ Coefficients $a$ and $U$ in the expansions \rf{superfacass} and \rf{abelFacExp} \label{CoefficientsAUFac}}
\baselineskip 12pt
\begin{tabular}{rrrr}
$n$ & ~~~ $~ a_{n}\phantom{0.1234567890123}$ & $~U_{n}\phantom{0.1234567890123}$& \rule{0pt}{14pt} \\
$ 2$&$ 0.7987318351724345 $&$-0.7987318351724345$& \rule{0pt}{12pt}\\ 
$ 3$&$ 0.5778809754764832 $&$ 0.6980641135593670$&\\ 
$ 4$&$ 0.3939788096629718 $&$-0.6339640557572815$&\\ 
$ 5$&$ 0.2575339580323327 $&$ 0.5884152357911399$&\\ 
$ 6$&$ 0.1629019581037053 $&$-0.5538887519936520$&\\ 
$ 7$&$ 0.1002824191713524 $&$ 0.5265479025985924$&\\ 
$ 8$&$ 0.0603184725913977 $&$-0.5041914604280215$&\\ 
$ 9$&$ 0.0355544582258062 $&$ 0.4854529800293392$&\\ 
$10$&$ 0.0205859954874424 $&$-0.4694346809094714$&
\end{tabular}
\end{center}
\end{table}
\vskip 2mm

For factorial, the increment $k\!>\!0$; expansion \rf{superfacass} gives good approximation $\tilde F(z)$ at
$-\Re(z)\gg 1$. For other $z$, I use the integer iterates; the superfunction
\be
F(z) = \lim_{n \rightarrow\infty}  \mathrm{Factorial}^n\!\Big(\tilde F(z\!-\!n) \Big) 
\eL{regufac}
For $\Re(z)\! \sim \!1$, it is sufficient to iterate expression  \rf{regufac} only few times, in order to get $\varepsilon$ of order of $0.1$ or less.
Then, some 15 terms in the expansion \rf{superfacass} provide of order of 14 correct significant figures of $F(z)$.
\vskip 2mm



Let $z_3$ be is real solution of equation $F(z_3)\!=\!3 $; the evaluation gives $z_3 \!\approx\! - 0.91938596545217788$ ; %есть вещественное решение уравнения 
then, I define
\be
\mathrm{SuFac}(z) = F(z_3\!+\!z) 
\eL{SuFac}
Function $\mathrm{SuFac}$ is also superfunction of factorial, and
$~\mathrm{SuFac}(0)\!=\!3~$.
\vskip 2mm

\begin{wrapfigure}{r}{246px}
\normalsize
\hskip 2pt
\sx{.29}{ \begin{picture}(602,1010)
\put(0,0){\includegraphics{SuperFacPlot}}
\put(372,978){\sx{4.5}{$y$}}
\put(372,890){\sx{4.5}{$9$}}
\put(372,790){\sx{4.5}{$8$}}
\put(372,690){\sx{4.5}{$7$}}
\put(372,590){\sx{4.5}{$6$}}
\put(372,490){\sx{4.5}{$5$}}
\put(372,390){\sx{4.5}{$4$}}
\put(372,290){\sx{4.5}{$3$}}
\put(372,190){\sx{4.5}{$2$}}
\put(372,90){\sx{4.5}{$1$}}
\put( 066,-42){\sx{4.5}{$-\!3$}}
\put(166,-42){\sx{4.5}{$-\!2$}}
\put(266,-42){\sx{4.5}{$-\!1$}}
\put(391,-42){\sx{4.5}{$0$}}
\put(491,-42){\sx{4.5}{$1$}}
\put(591,-42){\sx{4.5}{$2$}}
\put(691,-42){\sx{4.5}{$3$}}
\put(777,-42){\sx{4.5}{$x$}}
\put(532,440){\sx{5}{\rot{83}$y\!=\!\mathrm{SuFac}(x)$\ero}}
\put(740,500){\sx{5}{\rot{83}$y\!=\!\mathrm{Factorial}(x)$\ero}}
\put(850,0){\sx{3.1}{\rot{90}\url{https://mizugadro.mydns.jp/t/index.php/File:Sufact.png}\ero}}
\end{picture} }
\vskip 8pt
\caption{%\large 
$y\!=\!x!$ and
$y\!=\!\mathrm{SuFac}(x)$ by
\rf{regufac},\rf{SuFac}
%versus $x$
\label{SuperFacPlot}
}
\vskip -22pt
\end{wrapfigure}

I have displaced the argument of superfactorial, in order to let it have integer value at zero. This is smallest integer, that is still greater than the fixed point $L=2$.
\vskip 2mm

Graphic of function SuFac of the real argument is shown in figure 
\rf{SuperFacPlot},
$y\!=\!\mathrm{SuFac}(x)$. For comparison, in the same figure, factorial is shown,
$y\!=\!\mathrm{Factorial}(x)$.
Graphic of factorial passes through points with coordinates
(0,1), (1,1), (2,2), (3,6).
Graphic of the superfactorial passes through points with coordinates
(0,3), (1,6), (2,720); 
the last point is far away from the range of the figure.
\vskip 2mm

\begin{figure}[h!]
\hskip 28pt
\url{https://mizugadro.mydns.jp/t/images/d/d2/Superfactocomple1.png}
\vskip 8pt
\hskip -12mm
\sx{1.25}{ \begin{picture}(302,362)
\put(0,0){\ing{SuperFacMap}}
 \put(30,357){\sx{1.3}{$y$}}
 \put(30,317){\sx{1.3}{$6$}}
 \put(30,277){\sx{1.3}{$4$}}
 \put(30,237){\sx{1.3}{$2$}}
 \put(29,196){\sx{1.3}{$0$}}
 \put(20,156){\sx{1.3}{$-2$}}
 \put(20,116){\sx{1.3}{$-4$}}
 \put(20,76){\sx{1.3}{$-6$}}
 \put(20,36){\sx{1.3}{$-8$}}
 \put(70,30){\sx{1.3}{$-6$}}
 \put(110,30){\sx{1.3}{$-4$}}
 \put(150,30){\sx{1.3}{$-2$}}
 \put(198,30){\sx{1.3}{$0$}}
 \put(238,30){\sx{1.3}{$2$}}
 \put(278,30){\sx{1.3}{$4$}}
 \put(318,30){\sx{1.3}{$6$}}
 \put(354,30){\sx{1.3}{$x$}}
\put(50,344){\sx{1.3}{$u\!=\!2$}}
\put(50,306){\sx{1.3}{$v\!=\!0$}}
\put(50,255){\sx{1.3}{$u\!=\!2$}}
\put(50,204){\sx{1.3}{$v\!=\!0$}} %central
\put(50,152){\sx{1.3}{$u\!=\!2$}}
\put(50,100){\sx{1.3}{$v\!=\!0$}}
\put(50,049){\sx{1.3}{$u\!=\!2$}}
 % column<br>
\put(122,342){\sx{1.2}{$v\!=\!-0.2$}}
\put(135,314){\sx{1.2}{$u\!=\!1.8$}}
\put(252,314){\sx{1.2}{$u\!=\!1.2$}}
\put(136,265){\sx{1.2}{$v\!=\!0.2$}}
\put(125,210){\sx{1.2}{$u\!=\!2.2$}}
\put(125,130){\sx{1.2}{$v\!=\!-0.2$}}
\put(134,084){\sx{1.2}{$u\!=\!1.8$}}
\put(252,084){\sx{1.2}{$u\!=\!1.2$}}
\put(134,054){\sx{1.2}{$v\!=\!0.2$}}
\put(322,343){\sx{1.3}{$u\!=\!1$}}
\put(322,306){\sx{1.3}{$v\!=\!0$}}
\put(322,269){\sx{1.3}{$u\!=\!1$}}
\put(266,247){$u\!=\!0.8856031944$}
\put(332,231){\sx{1.3}{$v\!=\!0$}} %central
\put(329,164){\sx{1.3}{$v\!=\!0$}} %central
\put(322,137){\sx{1.3}{$u\!=\!1$}}
\put(322,100){\sx{1.3}{$v\!=\!0$}}
\put(322, 50){\sx{1.3}{$u\!=\!1$}}
 \end{picture}}
 
 \hskip -2pt
 \sx{.78}{ \begin{picture}(362,210)
 %\sx{.88}{ \begin{picture}(362,210)
 \put(0,0){\ing{SuperFacZoom}}
 
\put(4,216){\sx{1.8}{$y$}}
\put(4,194){\sx{1.8}{$2$}}
\put(4,095){\sx{1.8}{$1$}}
\put(097,3){\sx{1.8}{$1$}}
\put(197,3){\sx{1.8}{$2$}}
\put(297,3){\sx{1.8}{$3$}}
\put(397,3){\sx{1.8}{$4$}}
\put(497,3){\sx{1.8}{$5$}}
\put(536,3){\sx{1.8}{$x$}}

\put(360,210){\sx{1.8}{$u\!=\!0.8856031944$}}
\put(311,198){\sx{1.3}{\rot{-51}$u\!=\!0.8$\ero}}
\put(146,159){\sx{2}{ \rot{62}$u\!=\!1$\ero }}
\put(172,117){\sx{2}{ \rot{1}$u\!=\!0$\ero }}
\put(020,163){\sx{2}{ \rot{-37}$u\!=\!2$\ero }}
\put(018,053){\sx{2}{ \rot{ 22}$u\!=\!3$\ero }}
\put(051,06){\sx{2}{ \rot{ 53}$u\!=\!4$\ero }}

\put(082,183){\sx{2}{ \rot{2}$v\!=\!1$\ero }}
\put(081,083){\sx{2}{ \rot{46}$v\!=\!2$\ero }}
\put(306,148){\sx{2}{ \rot{0.}$v\!=\!0$\ero }}
\put(406,148){\sx{2}{ \rot{0.}$v\!=\!0$\ero }}
 \end{picture}
 }
\vskip -2pt
\hskip 26pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sfaczoo300.png}
\vskip 2pt
\caption{ \large
 %$u\!+\!\mathrm i v \!=\! \mathrm{SuperFactorial}(x\!+\!\mathrm i y)$
 $u\!+\!\mathrm i v \!=\! \mathrm{SuFac}(x\!+\!\mathrm i y)$
 %и зумин центральной части карты
 and the zoom-in of the central part
 \label{SuperFacMap}}
\ \vskip -8pt
\end{figure}
\baselineskip 18.8pt

Complex map of SuFac is shown in figure \ref{SuperFacMap}.
This function has period
\be
P=\frac{2\pi \mathrm i}{k} =2\pi \mathrm i \, \ln\!\big(3\!+\! 2\, \mathrm{Factorial}^\prime(0)\big)
\approx 10.253449681156%0279265772640691397 
\,\mathrm i
\ee 
This period is pure imaginary.
A little bit less than two periods fit the height of the upper map in figure \ref{SuperFacMap}.
\vskip 2mm

In the half-strips 
\be
x\!>\!2, ~\big |y + n |P|\big|<1, ~{\text{for integer}}~ n  
\eL{sufastrips}
$\mathrm{SuFac}(x+\mathrm i y)$ has countable set of singularities.
At the map, every negative integer value of $\Im(\mathrm{SuFac}(x\!+\!\mathrm i y))$, at the translation for unity along the real axis,
produces the singularity, due to the transfer equation \rf{Tfac}.
In order to show the structure of these singularities, the zoom-in %of the central part 
of the map is shown at the bottom of figure
\ref{SuperFacMap}.
Outside the half-strips \rf{sufastrips}, factorial is regular; it approaches to the fixed points $L$ of factorial: 
$L\!=\!2$ in the left hand side and 
$L\!=\!1$ in the right hand side of the map.
\vskip 2mm

The fast growth of factorial implies the slow growth of the inverse function, id est, growth of the abelfactorial.
I denote this function with symbol $\mathrm{AuFac}$.
Complex map of %the inverse function 
$\mathrm{AuFac}\!=\!\mathrm{SuFac}^{-1}$ is shown in figure 
\ref{AbelFactorialMap};
this function is considered in the next section. 

\begin{figure}[t]
\hskip -34pt
\sx{1.25}{ \begin{picture}(322,356)
\put(0,0){\ing{AbelFacMap}}
\put(30,357){\sx{1.3}{$y$}}
\put(30,317){\sx{1.3}{$6$}}
\put(30,277){\sx{1.3}{$4$}}
\put(30,237){\sx{1.3}{$2$}}
\put(29,196){\sx{1.3}{$0$}}
\put(20,156){\sx{1.3}{$-2$}}
\put(20,116){\sx{1.3}{$-4$}}
\put(20,76){\sx{1.3}{$-6$}}
\put(20,36){\sx{1.3}{$-8$}}
\put(70,30){\sx{1.3}{$-6$}}
\put(110,30){\sx{1.3}{$-4$}}
\put(150,30){\sx{1.3}{$-2$}}
\put(198,30){\sx{1.3}{$0$}}
\put(238,30){\sx{1.3}{$2$}}
\put(278,30){\sx{1.3}{$4$}}
\put(318,30){\sx{1.3}{$6$}}
\put(354,30){\sx{1.3}{$x$}}
\put(50,306){\sx{1.3}{$u\!=\!1.8$}}
\put(50,255){\sx{1.3}{$v\!=\!0.6$}}
\put(90,198){\sx{1.3}{\bf cut}} %central%
\put(50,150){\sx{1.3}{$v\!=\!-0.6$}}
\put(48,90){\sx{1.3}{$u\!=\!1.8$}}
\put(170,344){\sx{1.3}{$u\!=\!1.6$}}
\put(119,230){\sx{1.3}{$v\!=\!0.8$}}
\put(109,170){\sx{1.3}{$v\!=\!-\!0.8$}}
\put(144,090){\sx{1.3}{$u\!=\!1.6$}}
\put(94,068){\sx{1.3}{$v\!=\!-0.6$}}
\put(252,084){\sx{1.3}{$u\!=\!1.4$}}
\put(319,344){\sx{1.3}{$u\!=\!1.4$}}
\put(304,306){\sx{1.3}{$v\!=\!0.4$}}
\put(313,278){\sx{1.3}{$u\!=\!1.2$}}
\put(309,236){\sx{1.3}{$v\!=\!0.2$}}
\put(306,209){\sx{1.3}{$u\!=\!1$}}
\put(331,198){\sx{1.3}{$v\!=\!0$}} %central
\put(312,118){\sx{1.3}{$u\!=\!1.2$}}
\put(304,90){\sx{1.3}{$v\!=\!-0.4$}}
\put(320, 50){\sx{1.3}{$u\!=\!1.4$}}
\end{picture}}
\vskip -14mm
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:AbelFacMapT.jpg}
\vskip -6pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{AbelFactorial}(x\!+\!\mathrm i y)$. \iL{AbelFactorialMap} \label{File:AbelFacMapT.jpg}}
\vskip 1pt
\end{figure}

\section{Abelfactorial}

The asymptotic series for super factorial can be inverted. This gives the asymptotic expansion of 
abelfactorial (arcsuperfactorial) in vicinity of the fixed point $L\!=\!2$:
\be \!\!\!\!\!\!\!\!\!\!
\tilde G(z)=
\frac{1}{k} \ln\!\left(\sum_{n=1}^{N-1}U_{n}(z\!-\!2)^{n}+\mathcal{O}(z\!-\!2)^N\right)
\eL{abelFacExp}
Parameter $k$ has the same meaning, as in expansion \rf{superfacass}; its value is determined by \rf{fack}.
The first two coefficients $U$ are
\be
\!\!\!\!\!\!\!\! U_1 &=\!&1 \\
\!\!\!\!\!\!\!\! U_2 &=\!&\!-\frac{\pi^2+6\gamma^{2}-18\gamma+6}{12(3-5\gamma+2\gamma^{2})} \approx 0.7987318
\eL{UAFac}
The Mathematica routine {\bf InverseSeries} allows to calculate these coefficients analytically, although the expressions for the highest $U$ 
are a little bit long. The approximate values of these coefficients are shown in the right column of table 
\ref{CoefficientsAUFac}. 
\vskip 2mm

For argument close to 2, the truncated series in \rf{abelFacExp} provides the precise evaluation.
For other values, the repeated application of the recurrent formula
\be
\!\!\!\!\!G(z) = G\Big(\mathrm{ArcFactorial}(z)\Big)+1
\eL{AbelFacDrugaya}
is used, until  $|z\!-\!2|$ becomes small.
This algorithm determines the cut line of the abelfctorial AuFac; the cut goes from 2 to $-\infty$.
\vskip 2mm

The abelfunction $G$ of factorial can be defined with 
\be
G(z)=\lim_{n\rightarrow \infty} \tilde G(\mathrm{Factorial}^{-n}(z)+n
\eL{facG}
\JP{
While %$\mathrm {SuFac}(3)\!=\!\mathrm{Factorial}^z(3)$ и, соответственно, 
$\mathrm{SuFac}(0)\!=\!3$,
The inverse function, I call it AuFac
}%JP
then, $\mathrm{AuFac}\!=\!\mathrm{AbelFactorial}\!=\!\mathrm{SuFac}^{-1}$,
appears as
\be
\mathrm{AuFac}(z)=G(z)-z_3=G(z)-G(3)
\eL{AuFac}
where $z_3=G(3)\approx  -0.91938596545218$.
This expression automatically provides condition $\mathrm{AuFac}(3)\!=\!0$.
In such a way,
\be
\mathrm{AuFac}(z) \approx \mathrm{AuFac}_n(z) = \tilde G \Big(\mathrm{ArcFactorial}^n(z)\Big)+n-z_3
\label{abelfac3}
\ee

\JP{
Такая аппроксимация не должна создавать впечатления, что построено лишь приближенное решение. Точную формулу можно выразить через предел, то есть
\be
\mathrm{AuFac}(z) =\lim_{n\rightarrow \infty}  \mathrm{AuFac}_n(z)
\eL{abelfac3e}
При этом абельфакториал оказывается обратной функцией суперфакториала. 
В частности, такой алгоритм использован для построения рисунка \ref{AbelFactorialMap}. %и \ref{AbelFactorialPlot}.
}%JP
\vskip 2mm

\begin{wrapfigure}{r}{253px}
\normalsize
\sx{.318}{\begin{picture}(806,580)
\put(0,0){\includegraphics{AbelFacPlo}}
\put(9,595){\sx{4}{$y$}}
\put(9,503){\sx{4}{$2$}}
\put(9,403){\sx{4}{$1$}}
\put(9,303){\sx{4}{$0$}}
\put(-2,203){\sx{4}{$-\!1$}}
\put(-2,103){\sx{4}{$-\!2$}}
\put(093,282){\sx{4}{$1$}}
\put(193,282){\sx{4}{$2$}}
\put(293,282){\sx{4}{$3$}}
\put(393,282){\sx{4}{$4$}}
\put(493,282){\sx{4}{$5$}}
\put(593,282){\sx{4}{$6$}}
\put(693,282){\sx{4}{$7$}}
\put(782,282){\sx{4}{$x$}}
\put(330,524){\sx{5}{\rot{8.8}$y\!=\!\mathrm{ArcFactorial}(x)$\ero}}
\put(380,390){\sx{5}{\rot{10}$y\!=\!\mathrm{AuFac}(x)$\ero}}
\end{picture}}
\vskip -10pt
\sx{.79}{\url{https://mizugadro.mydns.jp/t/index.php/File:Aufact.png}}
\vskip -5pt
\caption{\large ArcFactorial and AuFac
\label{AbelFactorialPlot}}
\vskip -8pt
\end{wrapfigure}

Function AuFac is plotted
in figure
\ref{AbelFactorialPlot}. 
For comparison, graphic of ArcFactorial is also shown.
Graphic of ArcFactorial goes through points
(1,1), (2,2), (6,3), (24,4), 
although the last one is already out of range of the figure (and out of range of the page.
Graphic of AuFac passes through points
(3,0), (6,1), (720,2), 
and the last one is not only our of range of the page, but also out of room, were this Book is written, and the next point, where AuFac takes the integer value, id est, (720!, 3) is far away from the visible part of our Universe.
\vskip 2mm

Function AuFac grows very slowly. If the argument of this function represents some physical quantity (distance, mass, charge, number of atoms, etc.), measured in any reasonable units, then, there is no way to make this quantity so big, that AuFac of it reaches 3.
\vskip 2mm

While superfactorial and abelfactorial, id est, SuFac and AuFac, are already constructed and implemented, one can use them to evaluate the non-integer iterates of factorial. These iterates are matter of the next section.

\section{Iterates of factorial}

With super factorial and abelfactorial, the $n$th iterate of factorial can be written as follows:
\be\!\!\!\!\!\!\!\!
h(z)=\mathrm{Factorial}^n(z)=\mathrm{SuFac}\Big(n+\mathrm{AuFac}(z)\Big) 
\eL{Facc}

\begin{figure}[t]
\hskip 2pt
\sx{.42}{\begin{picture}(1000,1004)
\put(0,0){\ing{facit}}% 
\put(4,984){\sx{4}{$y$}}
\put(4,890){\sx{4}{$9$}}
\put(4,790){\sx{4}{$8$}}
\put(4,690){\sx{4}{$7$}}
\put(4,590){\sx{4}{$6$}}
\put(4,490){\sx{4}{$5$}}
\put(4,390){\sx{4}{$4$}}
\put(4,290){\sx{4}{$3$}}
\put(4,190){\sx{4}{$2$}}
\put(4, 90){\sx{4}{$1$}}
\put(92,4){\sx{4}{$1$}}
\put(192,4){\sx{4}{$2$}}
\put(292,4){\sx{4}{$3$}}
\put(392,4){\sx{4}{$4$}}
\put(492,4){\sx{4}{$5$}}
\put(592,4){\sx{4}{$6$}}
\put(692,4){\sx{4}{$7$}}
\put(792,4){\sx{4}{$8$}}
\put(892,4){\sx{4}{$9$}}
\put(982,4){\sx{4.1}{$x$}}
\put(212,830){\rot{89}\sx{5}{$n\!\rightarrow\!\infty$}\ero}
\put(262,824){\rot{87}\sx{5}{$n\!=\!2$}\ero}
\put(340,816){\rot{83}\sx{5}{$n\!=\!1$}\ero}
\put(454,806){\rot{78}\sx{5}{$n\!=\!0.5$}\ero}
\put(494,806){\rot{74}\sx{5}{$n\!=\!0.4$}\ero}
\put(544,806){\rot{68}\sx{5}{$n\!=\!0.3$}\ero}
\put(606,806){\rot{61}\sx{5}{$n\!=\!0.2$}\ero}
\put(698,806){\rot{54}\sx{5}{$n\!=\!0.1$}\ero}
\put(766,744){\rot{45}\sx{5.2}{$n\!=\!0$}\ero}
\put(770,638){\rot{34}\sx{5}{$n\!=\!-0.1$}\ero}
\put(770,556){\rot{27}\sx{5}{$n\!=\!-0.2$}\ero}
\put(770,494){\rot{21}\sx{5}{$n\!=\!-0.3$}\ero}
\put(770,450){\rot{15}\sx{5}{$n\!=\!-0.4$}\ero}
\put(770,414){\rot{11}\sx{5}{$n\!=\!-0.5$}\ero}
\put(770,311){\rot{4}\sx{5}{$n\!=\!-1$}\ero}
\put(770,232){\rot{1}\sx{5}{$n\!=\!-2$}\ero}
\put(770,183){\rot{.01}\sx{5}{$n\!\rightarrow\!-\infty$}\ero}
\end{picture}}%$\!\!\!\!$
\vskip 2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:FacitT.jpg}
\vskip -4pt
\caption{\large $y\!=\!\mathrm {Factorial}^n(x)$ versus $x$
for various $n$
\iL{facit}}
\vskip 2pt
\end{figure}

In this representation $n$ has no need to be integer.
Figure \ref{facit} shows graphic $y\!=\!\mathrm{Factorial}^n(x)$ versus $x$ for various real $n$.
Figure \ref{facit} should be compared to figures \ref{itelin125}, 
\ref{f1ziteplot},
\ref{c05},
\ref{c10},
\ref{c20}, \ref{IterPowPlot}, \ref{KellerPlot},
that show iterates of other functions, that can be expressed through the special functions, known at least since century 20.
There is no similar representation for the abelfactorial. I hope, in century 21, some of new superfunctions, including
SuFac, AuFac 
will be included in the handbooks on the special functions
and implemented as built-in routines in the programming languages; then, the difference between 
iterates of factorial and iterates of other functions, mentioned in the previous chapters, will be even less significant.
\vskip 2mm

\begin{figure}[h]
\vskip 8pt
\hskip -24pt
{\sx{1.22}{\begin{picture}(362,364)
\put(0,0){\ing{QFacMap}}
\put(30,357){\sx{1.3}{$y$}}
\put(30,317){\sx{1.3}{$6$}}
\put(30,277){\sx{1.3}{$4$}}
\put(30,237){\sx{1.3}{$2$}}
\put(29,196){\sx{1.3}{$0$}}
\put(20,156){\sx{1.3}{$-2$}}
\put(20,116){\sx{1.3}{$-4$}}
\put(20,76){\sx{1.3}{$-6$}}
\put(20,36){\sx{1.2}{$-8$}}
\put(70,30){\sx{1.1}{$-6$}}
\put(111,30){\sx{1.1}{$-4$}}
\put(151,30){\sx{1.1}{$-2$}}
\put(199,30){\sx{1.1}{$0$}}
\put(239,30){\sx{1.1}{$2$}}
\put(279,30){\sx{1.1}{$4$}}
\put(319,30){\sx{1.1}{$6$}}
\put(355,30){\sx{1.1}{$x$}}
\put(70,198){\sx{1.3}{\bf cut}} %central
\put(47,162){\sx{1.3}{\rot{72}$u\!=\!0$\ero}}
\put(99,161){\sx{1.3}{\rot{80}$v\!=\!0$\ero}}
\put(139,162){\sx{1.3}{\rot{89}$u\!=\!0$\ero}}
\put(164,204){\sx{1.3}{\rot{77}$v\!=\!0$\ero}}
\put(177,200){\sx{1.2}{\rot{56}$u\!=\!0$\ero}}
\put(82,116){\sx{1.1}{\rot{0.}$u\!=\!0.4$\ero}}
\put(86,102){\sx{1.1}{\rot{0.}$u\!=\!0.6$\ero}}
\put(137,128){\sx{1.1}{\rot{0.}$v\!=\!0.4$\ero}}
\put(332,252){\sx{1.3}{\rot{23}$u\!=\!8$\ero}}
\put(331,220){\sx{1.3}{\rot{-12}$v\!=\!8$\ero}}
\put(321,174){\sx{1.3}{\rot{9}$v\!=\!-8$\ero}}
\put(319,102){\sx{1.3}{\rot{-7}$u\!=\!-8$\ero}}
\end{picture}}}
\vskip -36pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:QFacMapT.jpg}
\vskip -6pt
\caption{\large  $u\!+\!\mathrm i v=\mathrm{Factorial}^{0.5}(x\!+\!\mathrm i y)$
\iL{QFacMap}}
\vskip -.2mm
\end{figure}

At $n\!=\!1/2$, formula \rf{Facc} gives the half iterate of factorial.
Map of function $h\!=\!\mathrm{Factorial}^{1/2}$ 
is shown in figure \ref{QFacMap}. 
This function can be interpreted as ``square root of factorial'' 
\sx{1.1}{$\sqrt{\,!\,}\,$}, 
used as logo of the Phys. Dep. of the Moscow State University, shown in figure
\ref{SqrtFactorialLogo}.
\vskip 2mm

Concept of square root of factorial caused confusions and discussions.
Some colleagues did not want to see difference between expiressions
$\mathrm{Factorial}^{1/2}(z)$ and
$\mathrm{Factorial}(z)^{1/2}$, 
interpreting 
$\sqrt{\mathrm{Factorial}}(z)$ 
as equivalent of 
$\sqrt{\mathrm{Factorial}(z)}$. 
In order to eliminate, en fin, this confusion, function
$h\!=\!\mathrm{Factorial}^{1/2}$ 
is considered here with more details.
\vskip 2mm

The key for the verification of interpretation of function
\sx{1.01}{$h=\sqrt{\,!\,}\,$},
is analysis of the range of validity of relation
\be
h(h(z))=\mathrm{Factorial}(z)=z!   
\eL{hh}
Map of function $h^2$ in the left hand side of equation \rf{hh} is shown in figure 
\ref{QQFacMap}. 
This figure should be compared to figure \ref{facmap}, 
that represents the complex map of factorial.
\vskip 2mm
\begin{figure}[t]
\sx{1.22}{\begin{picture}(362,362) \ax
\put(-20,-10){\includegraphics{QQFacMap}}
\put(182,263){\rot{42}\sx{1.6}{\bf cut}\ero}
\put(299,187){\sx{1.6}{$v\!=\!0$}}
\end{picture}}
\vskip -25pt
\hskip 20pt
\url{https://mizugadro.mydns.jp/t/index.php/File:QQFacMT.jpg}
\vskip -6pt
\caption{$u\!+\!\mathrm i v=h^2(x\!+\!\mathrm i y)$.
\iL{QQFacMap}}
\end{figure}

In the right hand side of figure \ref{QQFacMap}, complex map of the second iterate of function $h$, id est, $h\circ h=h^2$, 
coincides with the map of factorial, shown in figure  \ref{facmap}.
At least in the halfstrip $\Re(z)\!>\!1$, $|\Im(z)| \!\le\! 4$, relation \rf{hh} holds.
The resulting half iteration of factorial corresponds to the intuitive expectations about this function.
\vskip 2mm
Range of validity of relation  \rf{hh} is limited with cut lines, shown in figure \ref{QQFacMap}. Such cutlines are typical for non-integer iterates, if the superfunction cannot be represented through elementary function. 
\vskip 2mm

Factorial is not exotic, not an exception.
With superfunctions, one can build-up non-integer iterates for other holomorphic functions too.  More examples are considered in the following chapters below. 
\vskip 8pt

Many transfer functions can be treated in a way, similar to that of this chapter.
One of them, namely, the exponential to base $b=\sqrt{2}$, is considered in the next chapter.
\JP{
Нецелые итерации функций
расширяют инструментарий, например, для аппроксимации (фиттинга) различных неизвестных функций. К любой аппроксимирующей функции добавляется возможность итерировать её произвольное количество раз. 
\vskip 2mm

\begin{wrapfigure}{r}{208px}
\vskip -24pt
\sx{.27}{\begin{picture}(1013,600)
\put(-2,0){\ing{purgu}}
\end{picture}}
\vskip -10pt
\caption{\large
Что значит {\em гнать пургу}  \label{purgu}}
\vskip -24pt
\end{wrapfigure}
Расширение инструментария 
важно для экстраполяций. Удачная фитирующая функция имеет мало параметров, это улучшает предсказательную способность экстраполяции. Безграмотное (или жульническое) использование примитивных аппроксимаций с множеством параметров ведет к неверным результатам. Примером такой концепции является 
``глобальное потепление'' \footnote{ 
\url{https://mizugadro.mydns.jp/t/index.php/File:Purgu.png} Глобальное потепление \\
\url{https://rus.ruvr.ru/2009/12/18/3089576/} 18 декабря 2009, 18:22. Россия будет бороться с потеплением вне зависимости от результатов саммита. Дмитрий Медведев рассказал о лидирующей роли России в борьбе с глобальным потеплением.
 }, с помощью которого жулики гонят пургу (рис. \ref{purgu}) и пилят бюджет.
Я надеюсь, что нецелые итерации факториала не будут использоваться для такого мошенничества - хотя бы потому, что суперфункции позволят уменьшить число подгоночных параметров и ограничить возможности для всяческих артефактов в предсказаниях
- как умышленных, так и невольных.
\vskip 2mm

Факториал является быстро растущей функции с вещественной стационарной точкой. Стационарная точка 4 использована для построения суперфакториала, абельфакториала и нецелых итераций факториала. Ещё пример быстрорастущей передаточной функции с вещественной стационарной точкой рассмотрен в следующей главе.
}%JP
\chapter{~ Exponent to base sqrt(2) ~ \label{ChapterQ2}}
\baselineskip 19.4pt

\begin{figure}[h]

\sx{.333}{\begin{picture}(1202,700) %\put(10,10){\ing{ExpQ2plot}}
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\put(590,298){\sx{4.2}{$3$}}
\put(590,198){\sx{4.2}{$2$}}
\put(590,098){\sx{4.2}{$1$}} 
\put(080,-26){\sx{4}{$-5$}}
\put(180,-26){\sx{4}{$-4$}}
\put(281,-26){\sx{4}{$-3$}}
\put(381,-26){\sx{4}{$-2$}}
\put(482,-26){\sx{4}{$-\!1$}}
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\put(903.9,-26){\sx{4}{$3$}}
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\put(1104.1,-26){\sx{4}{$5$}}
\put(1192.2,-26){\sx{4.3}{$x$}}
\put(1034,490){\sx{5.6}{\rot{66}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}}
\put(1094,444){\sx{5.7}{\rot{45}$y\!=\!x$\ero}} 
\put(10,10){\ing{ExpQ2plot}}
\end{picture}}
\vskip 8pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ExpQ2plotT.png}
\vskip -4pt
\caption{\normalsize Exponent to base $b\!=\!\sqrt{2}$
\label{exq2plo}
\label{ExpQ2plot}
}
\vskip 4pt
\end{figure}

Exponent to base $\sqrt{2}$ happened to be first 
function treated with the regular iteration \cite{sqrt2}.
With this example (and for this function), the methods, described in chapter 6, were developed.
In this chapter, this exponent to this specific base is considered as transfer function $T$.
\vskip 2mm

Exponent to base $\sqrt{2}$ can be 
expressed
through the natural exponent:
\be\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!
T(z)=\exp_{\sqrt{2}}(z)=\exp\!\Big( \ln(\sqrt{2}) z\Big) = \exp\!\left(\frac{\ln(2)}{2}\, z\right)
\eL{Texpq2}
Explicit plot of this function is shown in figure \ref{exq2plo}. 
\vskip 2mm

For exponential to base $b\!=\!\sqrt{2}$, the fixed points $L\!=\!2$ and $L\!=\!4$ are 
natural numbers. 
In figure \ref{exq2plo}, they correspond to intersections of curve $y\!=\!T(x)$ with the straight line $y\!=\!x$.

\begin{figure} [t] %1.48
\hskip 18pt
\sx{2.5}{\begin{picture}(162,162) 
\put(1,1){\ing{ExpQ2map}} 
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\put(16,-4){\sx{.6}{$-6$}}
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\put(58,119){\sx{.6}{\rot{90}$v\!=\!0.4$\ero}}
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\put(39,24){\sx{.6}{\rot{90}$v\!=\!-0.2$\ero}}
\put(59,24){\sx{.6}{\rot{90}$v\!=\!-0.4$\ero}}
\put(84.6,26){\sx{.7}{\rot{90}$v\!=\!-1$\ero}}
\put(104.6,26){\sx{.7}{\rot{90}$v\!=\!-2$\ero}}
\end{picture}} 
\vskip 6pt
\hskip 32pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ExpQ2mapT.png}
\vskip -6pt
\caption{\large
$u\!+\!\mathrm i v=\exp_{\sqrt{2}}(x\!+\mathrm i y)$
\label{ExpQ2map}}
\end{figure}

\begin{figure} [h] 
\hskip 18pt
\sx{2.5}
{\begin{picture}(162,162)
\put(1,1){\ing{LogQ2map}}
\put(-3,159.9){\sx{.6}{$y$}}
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\put(-3,100){\sx{.6}{$2$}}
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\put(-8,60){\sx{.6}{$-2$}}
\put(-8,40){\sx{.6}{$-4$}}
\put(-8,20){\sx{.6}{$-6$}}
\put(-8, 0){\sx{.6}{$-8$}}
\put(-4,-4){\sx{.6}{$-8$}}
\put(16,-4){\sx{.6}{$-6$}}
\put(36,-4){\sx{.6}{$-4$}}
\put(56,-4){\sx{.6}{$-2$}}
\put(81,-4){\sx{.6}{$0$}}
\put(101,-4){\sx{.6}{$2$}}
\put(121,-4){\sx{.6}{$4$}}
\put(141,-4){\sx{.6}{$6$}}
\put(159.6,-4){\sx{.6}{$x$}}
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\put(129,118.6){\sx{.7}{\rot{40}$v\!=\!2$\ero}}
\put(138,100.2){\sx{.7}{\rot{20}$v\!=\!1$\ero}}
\put(004,084.2){\sx{.66}{$v\!=\!9$}}
\put(141,080.2){\sx{.7}{$v\!=\!0$}}
\put(003,076){\sx{.66}{$v\!=\!-9$}}
\put(136,061){\sx{.66}{\rot{-21}$v\!=\!-1$\ero}}
\put(125,044){\sx{.66}{\rot{-41}$v\!=\!-2$\ero}}
\put(109.4,032){\sx{.66}{\rot{-61}$v\!=\!-3$\ero}}
\put(80,051){\sx{.66}{\rot{11}$u\!=\!3$\ero}}
\put(83,039.6){\sx{.66}{\rot{11}$u\!=\!4$\ero}}
\put(86,023.3){\sx{.66}{\rot{11}$u\!=\!5$\ero}}
\put(90,000.2){\sx{.66}{\rot{11}$u\!=\!6$\ero}}
\end{picture}}
\vskip 10pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ExpQ2mapT.png}
\vskip -4pt
\caption{\large $u\!+\!\mathrm i v=\log_{\sqrt{2}}(x\!+\mathrm i y)$
\iL{LogQ2map}}
\vskip 4pt
\end{figure}

\JP{ Юля попросила меня в Книге идти от простого к сложному. Как уже упомянуто в начале, в жизни все часто происходит гораздо по-другому, то есть от сложного к простому. Случай натуральной экспоненты (то есть по основанию $\mathrm e$) сложнее, чем случай экспоненты по основанию $\sqrt{2}$), и именно для этого случая вещественно-голоморфная суперфункция (то есть суперэкспонента) была построена раньше. Однако я стараюсь следовать совету Юли хотя бы в Книге, и сперва рассказываю про более простой случай, а именно, про неограниченно растущую суперфункцию экспоненты по основанию $\sqrt{2}$.
Этот случай во многом похож на случай факториала; растущая суперфункция такой экспоненты строится по тем же формулам, что и для факториала, и, к тому же, ведет себя схожим образом.
}%JP
Complex map of exponent to base  $b\!=\!\sqrt{2}$ is shown in figure \ref{ExpQ2map}. 
This function is periodic; its period
\be\!\!\!\!\!\!\!\!
P=\frac{2 \,\pi\, \mathrm i}{ \ln(b) }=\frac{2 \,\pi\, \mathrm i}{ \ln\Big(\sqrt{2} \Big) } \approx 18.1294405673 \,\mathrm i
\eL{Q2P18}
is pure imaginary.
A little bit less than one period fits the height of map in figure \ref{ExpQ2map}.
\vskip 2mm

\JP{Чтобы проследить построение суперфункции для экспоненты по основанию $b\!=\!\sqrt{2}$, полезно вспомнить свойства такой  передаточной функции. Эти свойства рассмотрены ниже.
\vskip 2mm
Пусть передаточная функция 
\be \!\!\!
T(z)=b^z=(\sqrt{2})^z=\exp_{\sqrt{2}}(z)=\exp\Big( \frac{\ln(2)}{2} z\Big)
\eL{Texpq2}
при $b\!=\!\sqrt{2}$; это значение, по умолчанию, принято в этой секции. Обратная функция
\be 
T^{-1}(z)=\log_{\sqrt{2}}(z)=\log_b(z)=\frac{2}{\ln(2)} \ln(z)
\eL{Tm1q2}
Чтобы вычислять экспоненту, нужно знать свойства умножения и его обратной операции, то есть деления. Примерно так же, для того, чтобы вычислять итерации экспоненты и суперэкспоненту, нужно знать свойства экспоненты её обратной функции. %операции, то есть логарифма. 
Иначе может случиться, что после лекции об устройстве паровоза, курсант-машинист спросит, куда в паровозе запрягаются лошади, а после курса атомной физики - о том, как откачивают воздух из пространства между ядром и электронами.
}%JP
For the exponential to base $b\!=\!\sqrt{2}$, the inverse function is logarithm to base $b$; this logarithm can be interpreted as minus first iterate:
\be
T^{-1}(z)=\log_{\sqrt{2}}(z)=\log_b(z)=\frac{2}{\ln(2)} \ln(z)
\eL{logsqrt2}
Complex map of logarithm to base 
$b\!=\!\sqrt{2}$ 
is shown in figure
\ref{LogQ2map}. 
The dashed line marks the cut of the range of holomorphism; that cut runs from zero along the negative part of the real axis. The jump at the cut line is determined by the period 
\rf{Q2P18} of the exponent:
\be
\log_{\sqrt{2}}(x+\mathrm i o) -
\log_{\sqrt{2}}(x-\mathrm i o) = P \approx 18.1294405673 \, \mathrm i
\label{logq2jump}
\ee
for $x\!<\!0$.
At the map, the levels 
$v\!=\!\Im(\log_{\sqrt{2}}(x\!+\!\mathrm i y))\!=\!-9$ and
$v\!=\!\Im(\log_{\sqrt{2}}(x\!+\!\mathrm i y))\!=\!9$ 
are seen close to the negative part of the real axis.
\vskip 2mm

Fixed points $L\!=\!2$ and $L\!=\!4$ of the exponent are also fixed points of the logarithm;
these points are seen in figures
\ref{ExpQ2plot}, \ref{ExpQ2map} and \ref{LogQ2map}. 
\vskip 2mm

This chapter describes the regular iteration of exponential to base  $\sqrt{2}$ at the fixed point  $L\!=\!4$.
The next section describes the construction of the superfunction, id est, the growing superexponential to this base.

\section{Superfunction at fixed point $L\!=\!4$}
\baselineskip 16.4pt

\begin{wrapfigure}{r}{190px}
\normalsize
\vskip 2pt
\sx{.21}{
\begin{picture}(808,2002) 
\put(0,0){\ing{sqrt23uplo}} 
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\put(262,-40){\sx{4.6}{$-1$}}
\put(392,-40){\sx{4.6}{$0$}}
\put(492,-40){\sx{4.6}{$1$}}
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\put(692,-40){\sx{4.6}{$3$}}
\put(780,-40){\sx{4.6}{$x$}}
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\put(550,186){\sx{5.5}{\rot{43}$y\!=\!\exp_{\sqrt{2}}(x)$\ero}}
\put(868,650){\rot{90}\sx{4}{\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt23uplot.jpg}}\ero}
\end{picture} }
\vskip 2pt
\caption{\large ~ $y\!=\!\mathrm{SuExp}_{\sqrt{2},5}(x)$ 
by %equation
\rf{Ff45} and $y\!=\!\exp_{\sqrt{2}}(x)$ 
\label{sqrt23uplot}}
\vskip -22pt \large
\end{wrapfigure}

Chapter 6 describes the construction of iterates of a transfer function, that are regular in vicinity of its fixed point.
Here, that method is used for 
$T\!=\!\exp_{\sqrt{2}}\,$;
the superfunction is constructed, that exponentially approaches the fixed point
$L\!=\!4$ 
at large negative values of the real part of the argument, and grows to infinity at the large positive argument.
This function is denoted with 
$\mathrm{SuExp}_{\sqrt{2},5}$.
The last superscript at the name of the function indicates its value at zero, $\mathrm{SuExp}_{\sqrt{2},5}(0)=5$.
In figure
\ref{sqrt23uplot}, curve $y\!=\!\mathrm{SuExp}_{\sqrt{2},5}$ is compared to that of $y\!=\!\exp_{\sqrt{2}}(x)$.
Below I construct and describe function $\mathrm{SuExp}_{\sqrt{2},5}$.
\vskip 8pt

I use formula \rf{Lea2general}
of the asymptotic expansion of superfuction $f$:
\be \!\!\!\!\!\!\!
f(z)\!=\!\sum_{n=0}^{N-1}\, a_n\, \rme^{nkz} \!+\! O\big(\rme^{Nkz} \big)~
\label{q2Fz4} \ee
Here I assume, that 
$a_0\!=\!L\!=\!4$ and  $a_1\!=\!1$.
Coefficient $a_1$ could be chosen arbitrary, but then other coefficients depend on $a_1$. 
Alteration of $a_1$ is equivalent of scaling of parameter $\varepsilon$ and displacement of the argument of the superfunction.
Then, formula 
\rf{k38}
gives
\be\!\!\!\!\!\!
k\!=\!\ln\!\Big({\exp_{b}}'(4)\Big)\!=\!\ln\!\Big(4\ln(\sqrt{2})\Big)\!=\! \ln(2\ln(2)) \!\approx\!
0.32663425997828%098238
\label{q2k4} 
\ee
This value determines periodicity of the superfunction; its period
\be
P=P_4=\frac{2\,\pi\,\mathrm i}{k}= \frac{2 \pi \mathrm i }{\ln(2 \ln(2))}
\approx  19.236149042042854712 \, \mathrm i
\label{q2P4}
\ee

I use equations %\rf{k38},
\rf{a2.39},\rf{a2.40} or directly the transfer equation (which gives the same results)
to get coefficients $a$. 
While the transfer function is exponent to base $\sqrt{2}$, the transfer equation
can be written as follows:
\be
f(z\!+\!1)=\Big(\sqrt{2}\Big)^{f(z)} \eL{q2Transfereq}

\begin{table} [t]
\vskip -12pt
\caption{\large Coefficients $a$ and $U$ in expansions \rf{q2Fz4}, \rf{q2Gz4} 
\label{tableq2a4} }
\vskip -8pt
\begin{center}
\large
\begin{tabular}{rrr} $n$ & $a_{n} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~$ & $U_{n} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~$ \\ \input{tableq2L4}
\end{tabular}
\end{center}
\vskip -4pt
\end{table}
Substitution of expansion \rf{q2Fz4} into the transfer equation \rf{q2Transfereq} determines coefficients $a_m$ for $m\!>\!1$. In particular,
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
a_2\!&\!=\displaystyle ~ ~ \frac{\ln(2)/4}{1-2 \ln(2)} \!&\!\approx\! 0.448587431195261%22890 
~ ~
\label{q2a2}\\
a_3\!&\!=\displaystyle \!\frac
{\Big(1 \!-\! \ln(2)\Big) \, \ln(2)^2/12}
{1\!-\!2\ln(2)\!-\!4\ln(2)^2\!+\!8\ln(2)^3} 
\!& \!\approx\! 0.1903722467978067%5668 ~
~ ~
\label{q2L4a23}
\ee
Similar (but longer) expressions 
can be written
for the other coefficients $a$.
The first column in table \ref{tableq2a4} suggests the approximate values of coefficients  $a$ in expansion \rf{q2Fz4}.
\vskip 2mm

I use the truncation of the expansion \rf{q2Fz4}, taking into account  $N\!=\!20$ terms.
This gives the primary approximation
$\tilde f$ 
for superfunction
$f$:
\be
\tilde f(z)=\sum_{n=0}^{19}\, a_n\, \exp(nkz)
\eL{q2Fz4tilde20}  \begin{figure}[t]
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\put(20,103){\sx{.99}{$v\!=\!0$}}
\put(20,56){\sx{.99}{$u\!=\!4$}}
\put(20,07){\sx{.99}{$v\!=\!0$}}
\put(072,161){\rot{46}\sx{.8}{$v\!=\!0.2$}\ero}
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\put(067,114){\rot{46}\sx{.8}{$u\!=\!4.2$}\ero}
\put(084,107){\rot{52}\sx{.8}{$u\!=\!4.4$}\ero}
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\put(098, 11){\rot{74}\sx{.8}{$u\!=\!3.6$}\ero}
\put(113.8, 11){\rot{77}\sx{.8}{$u\!=\!3.4$}\ero}
\put(127, 11){\rot{80}\sx{.8}{$u\!=\!3.2$}\ero}
\put(140.2, 11){\rot{88}\sx{.99}{$u\!=\!3$}\ero}
\end{picture}}
\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2figf45bT.png}
\vskip -12pt
\caption{ \large
$~u\!+\! \mathrm i v = \tilde f ( x\!-\!1.11520724513161 + \mathrm i \, y )$ ;
\iL{sqrt2figf45b} ~ ~ ~  \iL{mapeq2F4}
}
\end{figure}
\vskip -18pt
Approximation $\tilde f$ is shown in figure \ref{sqrt2figf45b}. 
In order to simplify the comparison with other maps, the argument of function in this figure is displaced for the real constant
$x_{45}\!\approx\!- 1.11520724513161$; 
this constant provides approximate condition 
$f(x_{45})\approx 5$
and the exact equality
$\mathrm{SuEx}_{\sqrt{2},5}(0)=5$
for the superfunction constructed below and shown in figure \ref{sqrt2f45map}.
\vskip 6pt

The primary approximation \rf{q2Fz4tilde20} 
allows to plot the complex map of superfunction in the left hand side of the complex plane.
The residual at the substitution of the approximation \rf{q2Fz4tilde20} into the transfer equation \rf{q2Transfereq} 
becomes of order of rounding errors at
 $\Re(z)\!<\!-2$. %практически, 
The readers are invited to plot this residual by themselves (preferably), or to look at it in the right hand side of figure 4 of the original publication \cite{sqrt2}.
\vskip 2mm

\begin{figure}[t]
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 \put(-3,44){\sx{.6}{$-6$}}
 \put(-3,24){\sx{.6}{$-8$}}
 \put(-2,00){\sx{.6}{$-10$}}
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 \put(166,0){\sx{.6}{$6$}}
 \put(186,0){\sx{.6}{$8$}}
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 \put(20,199.4){\sx{.99}{$v\!=\!0$}}
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 \put(20,103){\sx{.99}{$v\!=\!0$}}
 \put(20,56){\sx{.99}{$u\!=\!4$}}
 \put(20,07){\sx{.99}{$v\!=\!0$}}
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 \put(067,114){\rot{46}\sx{.8}{$u\!=\!4.2$}\ero}
 \put(084,107){\rot{52}\sx{.8}{$u\!=\!4.4$}\ero}
 % 0 <br>
 \put(071,73){\rot{44}\sx{.8}{$v\!=\!-0.2$}\ero}
 \put(072.4, 11){\rot{66}\sx{.8}{$u\!=\!3.8$}\ero}
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 \end{picture} }
\vskip -2pt
\hskip 23pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2figf45eT.png}
\vskip -8pt
\caption{\large
$~u\!+\!\mathrm i v= \mathrm{SuExp}_{\sqrt{2},5}(x\!+\! \mathrm i y)$ by  \rf{Ff45} ~
\iL{sqrt2f45map}}
\end{figure}
\JP{ Левая часть рисунка \ref{sqrt2figf45b} показывает комплексную карту этой функции линиями $u=\rm const$ и линиями $v=\rm const$, где, как и на других картах,  вещественные $u$ и $v$ определяются соотношениями
\be
 u\!+\! \mathrm i v = \tilde f \Big( x\!-\!1.11520724513161 + \mathrm i \, y \Big)
\eL{sdvi1112}
Константа 1.11520724513161 выбрана для удобства сравнения этой карты с картой суперфункции $F_{45}$, для которой 
$F_{45}(0)\!=\!5$;  константа 5 есть наименьшее целое число, превышающее значение $L\!=\!4$.
\vskip 2mm
Методом регулярной итерации 
}%JP
Superfunction $f$ appears as limit
\be
f(z)= \lim_{n\rightarrow \infty}{\exp_b}^n\Big( \tilde f(z\!-\!n) \Big)
\eL{sqrt2regi45}
This limit does not depend on the number $N$ of terms in the primary approximation
\rf{q2Fz4tilde20}. 
Instead of 19 terms, one could choose another constant. However,  the more terms are taken into account, the faster does the limit converge.
Choosing $N\!=\!19$, I keep in mind the implementation complex double. For the approximation of $f$ with 14 significant figures, it is sufficient to choose
$n\!>\! \Re(z)\!+\!2$. 
\vskip 2mm

It is convenient, when at zero the superfunction takes integer value. I choose this value to be 5. This is smallest integer, that is still greater than the chosen fixed point $L\!=\!4$. I denote this superfunction with symbol $\mathrm{SuExp}_{\sqrt{2},5}$ and define it as follows: 
\be
\mathrm{SuExp}_{\sqrt{2},5}(z)=F(z)=f(x_{45}+z)
\eL{Ff45}
where $x_{45} \approx -1.11520724513161$ 
is real solution of equation
$f(x_{45})\!=\!0$.
Namely this constant is used for the displacement of the argument of function $\tilde f$ 
in figure \ref{mapeq2F4}
in order of simplify the comparison with map of function $\mathrm{SuExp}_{\sqrt{2},5}$ shown in figure 
\ref{sqrt2f45map}.
\vskip 6pt

In the left hand side of the complex plane, functions 
$z \!\mapsto\! \tilde f(z\!+\!x_{45})$ and $\mathrm{SuExp}_{\sqrt{2},5}$ 
asymptotically coincide. This gives way to the precise evaluation. In the whole complex plane, the superfunction can be approximated with any arbitrary precision; so,
it should be qualified as exact solution.
In such a way, the primary approximation $\tilde f$ provides the exact solution for the superfunction $\mathrm{SuExp}_{\sqrt{2},5}$ in the whole complex plane.
\vskip 2mm

For the transfer function $Т$ as exponential to base $b\!=\!\sqrt{2}$,
the regular iteration at the fixed point 
$L\!=\!4$ 
gives the function
$F\!=\!\mathrm{SuExp}_{\sqrt{2},5}$
that is holomorphic in the whole complex plane.
Graphic of this function is shown in figure
\ref{sqrt23uplot}.
Complex map of function $\mathrm{SuExp}_{\sqrt{2},5}$ 
is shown in figure 
\ref{sqrt2f45map}. 
\vskip 2mm

For function $F\!=\!\mathrm{SuExp}_{4,5}$, value at zero is chosen a smallest integer, which is still greater than $L\!=\!5$; so,
$F(0)\!=\!5$.
Then, function $F\!=\!\mathrm{SuExp}_{4,5}$ can be interpreted as iterate of exponent with initial value 5:
\be
F(z)=\mathrm{SuExp}_{\sqrt{2},5}(z)=\exp_{\sqrt{2},\mathrm u}^z(5) ~ ~ ~ ~ ~ 
\eL{expq2z5}
where subscript $\mathrm u$ indicates, that the regular iteration is built-up at the highest (``upper'') fixed point of the exponent. 

In order to evaluate the non-integer iterates, 
the abelfunction is also required; I mean,
$G\!=\!F^{-1}\!=\!\mathrm{AuExp}_{\sqrt{2},5}$. 
This abelfunction is described in the next section.


\section{Abelfunction at fixed point $L\!=\!4$}

Superfunction $F\!=\!\mathrm{SuExp}_{\sqrt{2},4}$ of the exponent to base $\sqrt{2}$ is descried in the previous section.
This section considers the inverse function, $G\!=\!\mathrm{AuExp}_{\sqrt{2},4}\!=\!F^{-1}\!=\!\mathrm{SuExp}_{\sqrt{2},4}^{-1}$.
Then, the combination of superfunction $F$ and abelfunction $G$ allows to evaluate the non-integer iterates of the transfer function
$T\!=\! \exp_{\sqrt{2}}$ by the general formula \rf{Tzx}.
\vskip 2mm

Expansion for the abelfunction $g\!=\!f^{-1}$ can be obtained, inverting the asymptotics 
\rf{q2Fz4}. 
This gives for the abelfunction $g$ expansion $\tilde g$ in the following form:
\be \!\!\!\!\!\!\!\!\!\!
\exp(k\tilde g(z))=\sum_{n=1}^{N-1} \, U_m \, (z\!-\!L)^n +  O(z\!-\!4)^N
\iL{q2Gz4} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
\label{expq2gass}
\ee
where $U_0\!=\!0$; this implies, that at the fixed point $L$, function $g$ becomes infinite.
Then, according to expansion \rf{q2Fz4}, we get $U_1\!=\!1$ and $U_2\!=\!-a_2$.
While coefficients $a$ are represented with exact expressions (with infinite precision), Matgematica or Maple can calculate of order of 10
coefficients $U$ in expansion  \rf{expq2gass}.
Routine, that perform this inversion, is called
{\bf InverseSeries}.
In particular,
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
U_2\!=\!&\!\displaystyle ~ ~ \frac{\ln(2)/4}{1-2\ln(2)} &\!\approx\! -0.4485874311952612289
\label{eq2U2} \\
 \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
U_3\!=\! &\! \displaystyle \! \frac{\Big(1\!+\!4\ln(2)\Big) \ln(2)^2/24}{1\!-\!2\ln(2)\!-\!4\ln(2)^2\!+\!8\ln(2)^3} \!\!&\! \approx 0.21208912005491969757
\label{eq2U3}
\ee
Approximate values of coefficients $U$ are shown in the right hand side column of table
\ref{tableq2a4}.
The same coefficients can be obtained also independently on the expansion of superfunction, by substitution to expansion
\be
\tilde  g(z)=\frac{1}{k} \,  \ln\!\left( \sum_{n=1}^{N-1} \, U_m \, (z\!-\!L)^n +  O(z\!-\!4)^N \right)
\eL{eq2ga}
into the Abel equation
\be
g(\exp_b(z))=g(z)+1
\eL{eq2gass}
In order to fit the with of the screen (at the authomatic computation) or the width of the page
(at the calculation with paper and pen), getting 
$\tilde g(z)$ 
by formyla
\rf{eq2ga} 
it worth to use the new variable 
$\zeta\!=\!z\!-\!L\!=\!z\!-\!4$. 
Then, first coefficients $U$ in expansion \rf{expq2gass} can be found even without computer.
\vskip 2mm

\begin{figure}[t]
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\put(6,5){\ing{sqrt2figL45e}}
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\put(-2,00){\sx{.6}{$-10$}}
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\end{picture}} 
\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2figL45eT.png}
\vskip -4pt
\caption{\large
$u\!+\!\mathrm i v=\mathrm{AuExp}_{\sqrt{2},5}(x\!+\!\mathrm i y)$ 
by formula
\rf{eq2G} 
\label{sqrt2figL45e}}
\end{figure}
Truncated 
expansion $\tilde g(z)$ approximates 
$g(z)$ at $|z\!-\!4|\!<\!2$. 
For other values, the iterates with the Abel equation \rf{eq2gass} can be used.
Abelfunction $G\!=\!F^{-1}$, shown in figure 
\ref{sqrt2figL45e}, appears as limit of these iterates:
\be \!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{AuExp}_{\sqrt{2},5}(z)=G(z)=\lim_{n\rightarrow \infty} \tilde g \Big(\log_b(z\!-\!n)\Big)+n + x_{45}
\eL{eq2G}
where $x_{45}\approx -1.11520724513161$ is the same constant,
that appears in the definition \rf{Ff45} of the superexponent $F=\mathrm{SuExp}_{\sqrt{2},5}$,
providing $F(0)\!=\!5$. % In equation \rf{eq2G}, this constant provides relation $G(5)\!=\!0$, which is supposed to hold for the inverse function of $F$.
\vskip 2mm

The inverse function of any non-trivial entire function has cut(s). The abelexponent $G\!=\!\mathrm{AuExp}_{\sqrt{2},5}$ is not exception.
The cut of the range of its holomorphism is shown in figure \ref{sqrt2figL45e}
with dashed line. 
This cut goes along the real axis from minus infinity to the branch point 4. The abelexponent has logarithmic singularity at this point.
This singularity corresponds to the exponential approach of the superesponent to the fixed point, as the real part of the argument goes to minus infinity.
\vskip 2mm

The readers are invited to plot the map of the region, where the relation below holds:
\be
\mathrm{SuExp}_{\sqrt{2},5}\Big(
\mathrm{AuExp}_{\sqrt{2},5}(z) \Big) = z
\eL{SuExpAuExpQ23}
and the map of range of validity of relation 
\be
\mathrm{AuExp}_{\sqrt{2},5}\Big(
\mathrm{SuExp}_{\sqrt{2},5}(z) \Big) = z
\ee
Especially for the exponent to base $\sqrt{2}$, the algorithms of evaluation of the growing superfunction and the abelfunction are loaded as\\
\url{https://mizugadro.mydns.jp/t/index.php/F45E.cin} and\\
\url{https://mizugadro.mydns.jp/t/index.php/F45L.cin} \\
They are implemented in C++ as complex double functions of complex double arguments. With these algorithms, the Reader can reproduce figures of this chapter, even without downloading generators of these figures.
\vskip 2mm

\section{Notations}
\baselineskip 18.4pt
The most of notations, used in this Book, are already introduced. This section suggests some kind of short overview of these notations.
$Т$ for the transfer function (practically, it is any holomorphic function, for which the superfunction and abelfunctions are considered): 
$F$ for the superfunction, and
$G\!=\!F^{-1}$ for the corresponding abelfunction.
These notations are convenient, while it is clear, which transfer function is kept in mind, and which of its superfunctions is denoted with $F$ and which of its inverse $G$ is assumed.
\vskip 2mm

However, these notations may cause confusions, if cited from other chapters. Functions $T$, $F$ and $G$ may have (and actually have) different meanings in different chapters. For this reason, in this chapter, I introduce also long names 
$\mathrm{SuExp}_{\sqrt{2},5}\!=\!F$ and
$\mathrm{AuExp}_{\sqrt{2},5}\!=\!G$.
The names above have simple mnemonics. The first letter indicates, that it refers to Superfunction or to Abelfunction.
This letter is capitalised, following the tradition of language Mathematica.
The second letter indicates, that the biggest, ``upper'' fixed point of the transfer function is used.
This have sense, if the transfer function $T$ is real-holomorphic, and has real fixed point(s), and one of them can be qualified as maximal, upper. This takes place for the exponent to base $\sqrt{2}$.
The following three characters refer to the name of the transfer function, id est, exp; again, its first letter is capitalised.
The subscript indicates the base of this  exponent, $b\!=\!\sqrt{2}$ and value of this superexponent at zero;
$F(0)\!=\!5$ and $G(5)\!=\!0$.
\vskip 2mm


\section{Iterates of exponent to base $b=\sqrt{2}$}
\baselineskip 18.4pt
With superfunction $F\!=\!\mathrm{SuExp}_{\sqrt{2},5}$ and abefunction $G\!=\! \mathrm{AuExp}_{\sqrt{2},5}$,
the $n$th iterate of transfer function $T\!=\!\exp_{\sqrt{2}}$ can be expressed as follows:
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
T^n(z) = \exp_{\sqrt{2},\mathrm u}^{~n}(z) =
\mathrm{SuExp}_{\sqrt{2},5} \Big( n \!+\! \mathrm{AuExp}_{\sqrt{2},5}(z)\Big)
\eL{q2Tn5}
This formula is valid at least for $\Re(z)\!>\!4$. If one choose the upper (or lower) border at the cut of abelfuntion $G\!=\! \mathrm{AuExp}_{\sqrt{2},5}$ between 2 and 4, the resulting iterate $T^n(z)$ is holomorphic in the whole compilex plane except the cut along line $z\!\le\!2$. 
In figure \ref{iterEq2plot},  this $T^n(x)$ is shown versus $x$ for various real values of $n$.
These iterates are defined at least for $x\!>\!2$. For integer $n$, the curves can be extended also to $x\!\le \!2$.
\vskip 2mm
\begin{figure}[t!]
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\put(1,1){\ing{IterEq2plot}} 
\put(11,1384){\sx{4.4}{$y$}} 
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\put(11,890){\sx{4}{$9$}} 
\put(11,790){\sx{4}{$8$}} 
\put(11,690){\sx{4}{$7$}} 
\put(11,590){\sx{4}{$6$}} 
\put(11,490){\sx{4}{$5$}} 
\put(11,390){\sx{4}{$4$}} 
\put(11,290){\sx{4}{$3$}} 
\put(11,190){\sx{4}{$2$}} 
\put(11,090){\sx{4}{$1$}} 
\put(91,6){\sx{4}{$1$}} 
\put(191,6){\sx{4}{$2$}} 
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\put(391,6){\sx{4}{$4$}} 
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\put(592,6){\sx{4}{$6$}} 
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\put(794,6){\sx{4}{$8$}} 
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\put(1282,6){\sx{4}{$13$}} 
\put(1380,6){\sx{4.4}{$x$}} 
\put(416,1158){\sx{5}{\rot{90}$n\!\rightarrow \! \infty$\ero}} 
\put(518,1250){\sx{5}{\rot{88}$n\!=\!3$\ero}}
\put(590,1250){\sx{5}{\rot{84}$n\!=\!2$\ero}}
\put(750,1250){\sx{5}{\rot{78}$n\!=\!1$\ero}}
\put(800,1240){\sx{5}{\rot{74}$n\!=\!0.8$\ero}}
\put(872,1240){\sx{5}{\rot{71}$n\!=\!0.6$\ero}}
\put(961,1240){\sx{5}{\rot{64}$n\!=\!0.4$\ero}}
\put(1084,1240){\sx{5}{\rot{54}$n\!=\!0.2$\ero}}
\put(1172,1152){\sx{5.5}{\rot{44}$n\!=\!0$\ero}}
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\put(1234, 558){\sx{5}{\rot{4}$n\!=\!-2$\ero}}
\put(1234, 492){\sx{5}{\rot{2}$n\!=\!-3$\ero}}
\put(694, 1032){\sx{6.6}{\rot{72}$y\!=\!b^x$\ero}}
\put(870,610){\sx{6}{\rot{16}$y\!=\!\log_b(x)$\ero}}
\put(825,510){\sx{6}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}}
\put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}}
\put(872,852){\sx{6}{\rot{44}$y\!=\!x$\ero}} 
\put(1180,374){\sx{5}{\rot{0.1}$n\!\rightarrow\!-\infty$\ero}} 
\end{picture}}
\vskip -2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:IterEq2plotT.jpg}
\vskip -8pt
\caption{\large
$y\!=\!\exp_{\sqrt{2},\mathrm u}^{~ n}(x)$
by 
\rf{q2Tn5}
for various $n$. 
\iL{iterEq2plot}}
\end{figure}

For $n\!=\!1/2$, the
complex map of function $T^n(z)$ by \rf{q2Tn5} is shown in figure 
\ref{iterEq2map}. 
This is half iterate of exponential to base $\sqrt{2}$:
\begin{figure}[t]
\sx{2}{ \begin{picture}(202,202) 
\put(6,5){\ing{esqrt2ite12map}}
\put(2,203.4){\sx{.7}{$y$}}
\put(2,184){\sx{.6}{$8$}}
\put(2,164){\sx{.6}{$6$}}
\put(2,144){\sx{.6}{$4$}}
\put(2,124){\sx{.6}{$2$}}
\put(2,104){\sx{.6}{$0$}}
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\put(-3,44){\sx{.6}{$-6$}}
\put(-3,24){\sx{.6}{$-8$}}
\put(-2,-1){\sx{.7}{$-\!10$}}
\put( 22,-1){\sx{.7}{$-8$}}
\put( 42,-1){\sx{.7}{$-6$}}
\put( 62,-1){\sx{.7}{$-4$}}
\put( 82,-1){\sx{.7}{$-2$}}
\put(106,-1){\sx{.7}{$0$}}
\put(126,-1){\sx{.7}{$2$}}
\put(146,-1){\sx{.7}{$4$}}
\put(166,-1){\sx{.7}{$6$}}
\put(186,-1){\sx{.7}{$8$}}
\put(204,-1){\sx{.7}{$x$}}
\put(056,103.5){\sx{.99}{\bf cut}}
\put(179,135.6){\sx{.9}{\rot{-19}$v\!=\!6$\ero}}
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\end{picture}}
\vskip 1pt
\hskip 20pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Esqrt2ite12mapT80.jpg}
\vskip -4pt
\caption{\large
$u\!+\!\mathrm i v\!=\!T^{1/2}(x\!+\!\mathrm i y)$ 
by formula
\rf{Eq2T12}.
\iL{iterEq2map}}
\vskip 8mm
\end{figure}
\be \!\!\!\!\!\!\!
T^{1/2}(z)
\!=\!\exp_{\sqrt{2},\mathrm u}^{1/2}
\!=\!F\!\left(\!\frac{1}{2}\!+\!G(z)\!\right)
\!=\!
\mathrm{SuExp}_{\!\sqrt{2},5}\!\left(\!\frac{1}{2} \!+\! \mathrm{AuExp}_{\!\sqrt{2},5}(z)\!\right)
\label{Eq2T12}
\ee
I remind, that symbol ``u'' in the subscript indicates, that for the regular iterate, the highest (``upper'') fixed point is used as asymptotics of superfunction at infinity.
\vskip 2mm

Abelfunction $G$, built up at the fixed point $L\!=\!4$, has logarithmic singularity at this point, and the corresponding cut;
the jump 
at this cut is constant; and this constant is just period of the superfunction $F$.
For this reason, the iterates, even non-initeger, and in particular, the half iterate by equation \rf{Eq2T12},
are regular at the point $L\!=\!4$. However, the non-integer iterates have branch point at another fixed point of the 
transfer function, namely, $L\!=\!2$. In figure \ref{iterEq2map}, the corresponding cut is marked with dashed line.
\vskip 2mm


\section{Intermediate finish}
\baselineskip 19pt

\begin{wrapfigure}{r}{190pt}
\sx{.33}{\begin{picture}(400,420)
\put(50,-20){\ing{highmount}}
\end{picture}}
\caption{View from the saddle point \label{highmount}}
\vskip -34pt
\end{wrapfigure}

Several results and tools for superfunctions, abelfunctions and non-integer iterates are already presented above.
Here I make a view on the way already past and announce, what will be in the future way,
as it is shown in figure \ref{highmount}. 
\vskip 2mm

This chapter, above, deals with transfer function $T(z)\!=\!\big(\sqrt{2}\big)^z\!=\!\exp_{\sqrt{2}}(z)$; superfunction $F$, abelfunction $G$ and iterates of the transfer function $T$ are described:
\be
F&=&\mathrm{SuExp}_{\sqrt{2},5}\\
G&=&\mathrm{AuExp}_{\sqrt{2},5}\\
\!\!\!\!\!\!\!\!\!\!
T^{n}(z)&=&F(n+G(z))=\exp_{\sqrt{2},\mathrm u}^{~n}(z) ~ ~ ~ ~ ~ ~ \iL{expbc}\label{expbn}
\ee
Iterates by equation \rf{expbc} are shown in figure \ref{iterEq2plot}. 
These iterates look similar to iterates of other growing functions, in particular, those shown in figures
\ref{IterPowPlotA} for iterates of the power function
and \ref{facit} for iterates of factorial.
One may expect, at least for the real-holomorphic growing functions, the similar iterates can be constructed in the similar way, even if the superfunction and the abelfunction cannot be expressed through the special functions described in the textbooks of century 20. I mean, with the regular iteration, 
\vskip 2mm



With the examples above, the Reader already understands, how can one iterate a holomorphic function in such a way, that the number to iterates has no need to be integer.
These iterates greatly extends the tools, available for scientific description of various processes, for approximation (fitting) of physical (and, perhaps, not only physical) dependences.
The non-integer iterates of exponential can be used, and, in particular, those to base $\sqrt{2}$. The example of such a non-integer iterates is shown in figure \ref{iterEq2map}. I believe, the Reader can easy plot other examples of the iterates of the exponential to this base, and also for some other values of base. 



\JP{Путешественник или восходитель, вскарабкавшийся на хребет или каменистый перевал, и чуток оборзевший от тяжёлого подъема, может обозреть дальнейший путь. Так же Читатель (а также и автор), прочтя, как получилось изложение регулярной итерации, может обозреть, какие ещё случаи итераций можно (и нужно) рассмотреть. Такое обозрение предлагается ниже.
\vskip 2mm
В этой секции представлен своего рода ``промежуточный финиш''. То есть обзор нескольких глав Книги.}%JP
\vskip 2mm

The results presented above may cause impression, that they finish the investigation of superfunctions, abelfunctions and non-integer interates,
and that, in future, one needs just to apply the regular iteration to various special functions.
Actually, 
it is not a case.
\vskip 2mm

For some functions, the regular iterates with asymptotic \rf{Lea2general},\rf{Lea2} cannot be constructed, as the equations for the coefficients of the expansion have no solution. One of these cases is considered in the next chapter and qualified as ``exotic''.
\vskip 2mm

\JP{
\begin{wrapfigure}[14]{right}{268px}
\sx{.36}{\begin{picture}(500,390)
\put(0,0){\ing{stone}}
\end{picture}}
\vskip -8pt
\caption{Может ли Бог создать камень такой большой, что сам не сможет сдвинуть его с места? 
\label{stone}} \vskip -2pt
\end{wrapfigure} }%JP

\chapter{~ Exotic iteration and exp to base $\mathrm e^{1/ \mathrm e}$}
\baselineskip 19pt
\begin{figure}[h]
\normalsize
\hskip 8pt
\sx{.342}{\begin{picture}(1202,770)
\put(590,792){\sx{4.2}{$y$}}
\put(590,698){\sx{4.2}{$7$}} 
\put(590,598){\sx{4.2}{$6$}}
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\put(590,398){\sx{4.2}{$4$}}
\put(590,298){\sx{4.2}{$3$}}
\put(620,274){\sx{4.2}{$\mathrm e$}}
\put(590,198){\sx{4.2}{$2$}}
\put(590,098){\sx{4.2}{$1$}}
\put(080,-22){\sx{4}{$-5$}}
\put(180,-22){\sx{4}{$-4$}}
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\put(703.7,-22){\sx{4}{$1$}}
\put(803.8,-22){\sx{4}{$2$}}
\put(877.,16){\sx{4}{$\mathrm e$}}
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\put(1104.1,-22){\sx{4}{$5$}}
\put(1196,-22){\sx{4.3}{$x$}}
\put(1118,678){\sx{4.5}{\rot{69}$y\!=\!\eta^x$\ero}}
\put(1130,550){\sx{4}{\rot{61}$y\!=\!(\sqrt{2})^x$\ero}}
\put(1134,488){\sx{5}{\rot{45.1}$y\!=\!x$\ero}}
\put(10,10){\ing{expe1eplot}}
%\put(-6,0){\sx{2.4}{\rot{90} \url{https://mizugadro.mydns.jp/t/index.php/File:Expe1eplotT.jpg} \ero}}
\put(-6,0){\rot{90} \sx{2}{\url{https://mizugadro.mydns.jp/t/index.php/File:Expe1eplotT.jpg}} \ero}
\end{picture}}
\caption{\large $y\!=\! b^x$
for
$~b\!=\!\eta\!=\!\mathrm e^{1/\mathrm e}\!\approx\! 1.44466786~$ 
and
$~b\!=\!\sqrt{2}\!\approx\!
1.41421356$  
\label{expe1eplot}}  
\vskip -4pt
\end{figure}

The regular iteration by \rf{Lea2general}-\rf{regi}
looks as general method of construction of superfunctions. However, in some cases, it cannot be applied as is.
 In particular, the asymptotic expansion
 \rf{Lea2general} 
becomes invalid, if the derivative of the transfer function at the fixed point is unity:
\be
T(L)=L~&,&~~T'(L)=1
\eL{Tprimeu}
In this case, expression $T'\!-\!1=T'(L)\!-\!1$ 
in denominator of fraction in the right hand side of equation \rf{a2.39} 
becomes zero; and the coefficients in expansion \rf{Lea2general} loss their meaning.
Exponential to base $b\!=
\!\mathrm e^{1/\mathrm e}\!\approx\! 1.44466786$ is an example of such an exotic transfer function.
It is shown in figure \ref{expe1eplot} and considered in this chapter.
\vskip 8pt

\section{Exotic iterate}

\begin{wrapfigure}{r}{120pt}
\sx{.4}{\begin{picture}(220,160)
\put(0,0){\ing{exo1}}
\end{picture}}
\vskip -8pt
\caption{Exotics 
\label{exoti}}
\end{wrapfigure}

The coincidence, that the derivative in equation \rf{Tprimeu} becomes unity, can be qualified as exotic. 
This determines the title of the chapter and that if this section.
Condition \rf{Tprimeu} is not only exotic possible, (see figure \ref{exoti}), but, first, I consider %namely
case \rf{Tprimeu}.
\vskip 2mm

The general formulas of chapter 6 fail for the special case $T^{\prime}(L)\!=\!1$,
and Henryk Trappmann expected, that for this case the precise evaluation of superfunction is very difficult, if at all.
In order to convince him, that it is doable, we had to write the paper \cite{e1e}. Part of that publication is described below.


This section considers the case, when the derivative of the transfer function $T$ in the fixed point $L$ is unity.
Then, formulas of section 6 for the regular iteration cannot be applied as is.

\vskip 2mm
Calculation of iterates of a function is simpler, if its fixed point is zero.
If the fixed point $L$  of transfer function $T$ is not zero, then, the superfunction $F$ can be represented in the following form:
\be
F(z)=f(z)+L
\eL{FfL}
Then, for function $f$, we get
\be\!\!\!\!\!\!\!\!\!\!\!\!
f(z\!+\!1)=F(z\!+\!1)-L=T(F(z))\!-\!L=T(L\!+\!f(z))-L
\label{FfL1}
\ee
We may define 
\be
T_{\mathrm {new}}(z)= T(L+z)-L
\eL{TnewTLzL}
and interpret this $T_{\mathrm {new}}$ as new transfer function, and $f$ is superfunction for it.
Below, the subscript $_{\mathrm {new}}$ is omitted. This is equivalent to assumption
\be
T(0)=0
\ee
\vskip 2mm

Let the Taylor expansion of the transfer function have the following form:
\be
T(z)= z+ v z^2 + w z^3 + ..
\eL{TExpan}
where $v\!\ne \!0$.
For such a transfer function, it is difficult (and, perhaps, impossible) to built-up any superfunction, that exponentially approaches zero at infinity.
But it is possible to construct the superfunction, that decays, roughly, as the inverse proportional function. Let 
\be
f(z)= \frac{a}{z}+\frac{b \ell}{z^2}+\frac{\alpha \ell^2\!+\!\beta \ell\!+\!\gamma}{z^3} + ..
\eL{FExpan}
where 
$a$, $b$, $\alpha$, $\beta$ and $\gamma$ are constants, and $\ell\!=\!\ln(z)$. 
Below I assume, that coefficients $v$ and $w$ of the expansion \rf{TExpan} are known, and show, how to calculate coefficients $a$ and $b$ 
of the asymptotic \rf{FExpan}.
\vskip 2mm

For the displacement of the argument $z\mapsto z\!+\!1$, 
expressions in the asymptotic representation \rf{FExpan}
are transformed in the following way:

\be
\frac{1}{z}&\mapsto& \frac{1}{z\!+\!1}=\frac{1}{z}\left(1+\frac{1}{z}\right)^{\!-1}= \frac{1}{z}- \frac{1}{z^2}+ \frac{1}{z^3}+ ..\\
\frac{1}{z^2}&\mapsto& \frac{1}{(z\!+\!1)^2}=\frac{1}{z^2}\left(1+\frac{1}{z}\right)^{\!-2}= \frac{1}{z^2}- \frac{2}{z^3}+ \frac{3}{z^4}+ ..\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!   \rule{0pt}{28pt}
\ell\!=\!\ln(z) &\mapsto& \ln(z\!+\!1)=\ln\!\left( z \cdot\left (1+\frac{1}{z} \right) \right) \nonumber\\
&& ~ ~ ~ ~=\ln(z)+\ln\!\left (1+\frac{1}{z} \right)  %\nonumber\\ && 
=  \ell + \frac{1}{z}- \frac{1}{2z^2}+ ..
\ee
Using these preparations, the left hand side of the transfer equation
\be
f(z\!+\!1)=T(f(z))
\eL{transff}
can be written as follows:
\be \!\!\!\!\!\!\!\!\!\!\!
f(z\!+\!1)\!=\!&\!\! \displaystyle \frac{a}{z}-\frac{a}{z^2}+\frac{a}{z^3}+\frac{b \ell+b/z}{z^2}\left(1-\frac{2}{z}\right)+ \frac{\alpha \ell^2\!+\!\beta \ell\!+\!\gamma}{z^3} + .. ~ ~ ~ ~ ~ ~ \nonumber\\
=\!&\!\!\!\!\!\!\!\! \displaystyle  \frac{a}{z}+\frac{1}{z^2} \Big(\! -\!a\! +\! b \ell \Big)+\frac{1}{z^3} \Big(\!a\!+\!b - 2b\ell + \alpha \ell^2\!+\!\beta \ell\!+\!\gamma  \Big) + .. 
\label{fplus1ttp}
\ee
The right hand side of equation \rf{transff} becomes
\be \!
T(f(z))= %\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
 \displaystyle \frac{a}{z}+\frac{ b \ell}{z^2}+ \frac{  \alpha \ell^2\!+\!\beta \ell\!+\!\gamma }{z^3}+ v\cdot\left(\frac{a}{z}+\frac{b\ell}{z^2}+..\right)^2 +
w\left(\frac a z +..\right)^3
  .. ~   \nonumber
\ee 
\vskip -22pt
\be
\!=\!
\frac{a}{z} + \frac{1}{z^2}\Big( b \ell \!+\!v a^2 \Big) + \frac{1}{z^3}\Big( \alpha \ell^2\!+\!\beta \ell\!+\!\gamma\! +\! 2 v a b \ell \!+\! w a^3 \Big)+\!.. 
~ ~ ~ ~ ~ ~ ~ 
\label{fplus1tt}
\ee
We should equalize the right hand sites of equations \rf{fplus1ttp} and \rf{fplus1tt}.

Coefficients at  $\frac{1}{z}$ coincide automatically.
Coefficients at  $\frac{1}{z^2}$ gives
\be
 -a  = va^2
\eL{ava2}
Equalisation of coefficients at $\frac{1}{z^3}$ gives
\be
 a+b-2b\ell  = 2 v a b \ell + w a^3
\eL{f1tttttt}
Variable $\ell=\ln(z)$ depends on $z$, but equation 
\rf{f1tttttt}
should be valid for various $z$. So, we get the two equations:
\be
-2b  = 2 v a b 
\eL{again}
which, at $b\!\ne\! 0$, gives the same, as \rf{ava2}, and also
\be
a+b=w a^3
\eL{f1tttt}
Solving these two equations, I get
\be
a=\frac{-1}{v} ~ ~ ~, ~ ~ ~ ~ b=\frac{-w}{v^3}+\frac{1}{v} 
\eL{avbwv}

One can add more terms in the expansion \rf{FExpan}, and, in the similar way, 
calculate coefficients $\alpha$, $\beta$, $\gamma$ and even higher, assuming, that the expansion 
\rf{TExpan} of the transfer function is known. 
In such  away, the coefficients of the asymptotic expansion
\rf{FExpan} of the superfunction $f$
are determined by the coefficients of expansion of transfer function $T$.
\vskip 2mm

The asymptotic solution  \rf{FExpan} allows evaluation of the transfer function at large $z$.
Then, as in the case of the regular iteration, the asymptotic solution can be extended 
to the whole complex plane (except the cut lines) with the transfer equaltion, applying one of the two formulas below:
\be
f(z)=T^n\Big(f(z-n)\Big)
\eL{iterA}
or
\be
f(z)=T^{-n}\Big(f(z+n)\Big)
\eL{iterB}
in order to make the argument of the superfunction large, bringing it to the range, where the truncated asymptotic expansion
\rf{FExpan} provides the required precision.
The choice of one of the formulas 
\rf{iterA} or \rf{iterB}
determines the position of the cut lines of the resulting superfunction.
The direction of the cut lines can be changed also with replacement of 
$\ell=\ln(z)$ to $\ell=\ln(-z)$.
In order to get unique solution, we need to choose (and postulate) the asymptotic behavior of the function we want to construct.
In the following section, the method above is applied to exponential to base $\mathrm e^{1/\mathrm e}$, and then, in the following chapters, to other functions.
\vskip 2mm

In principle, even with two terms in equation \rf{FExpan}, one can get the camera-ready maps of the superfunction and related functions. Perhaps, this is sufficient for the application in physics and other sciences, where the precision of measurements is usually less than 14 decimal digits. 
The precision can be infinitely improved with modification of the argument of the primary approximation $f$ with equations \rf{iterA} or \rf{iterB}, increasing the number $n$ of iterates.
\vskip 2mm

However, for the numerical tests, it worth to calculate several coefficients of expansion of superfunction.
This reduces the time of evaluation and improves the precision.
For various transfer functions, that are holomorphic in vicinity of the fixed point and have expansion beginning with 
\rf{TExpan}, some tens of coefficients of the expansion \rf{FExpan} can be calculated.
Examples of the numerical tests for such expansions are presented below.

\JP{
\section{Non-zero fixed point}
\baselineskip 19pt

Формулы предыдущей секции  написаны для передаточной функции, стационарнарной точкой которой является ноль, и производная в этой стационарной функции в нуле равна единице.
При этом упомянуто, что эти формулы допускают обобщение на случай, когда стационарная точка  не равна нулю. Такой случай рассмотрен ниже.
\vskip 2mm

Допустим, требуется построить суперфункцию $\mathcal F$ для передаточной функции $\mathcal T$, которая на бесконечности стремится к её стационарной точке $\mathcal L$, 
то есть
$\mathcal T(\mathcal L)=\mathcal L$,
и при этом $\mathcal T^\prime(\mathcal L)=1$.
\vskip 2mm

Пусть новая передаточная функция $T$ связана с исходной передаточной функцией $\mathcal T$ соотношением
\be
T(z)=\mathcal T(z+\mathcal L) -L
\eL{TzTzL}
Если $\mathcal L$ есть стационарная точка передаточной  функции $\mathcal T$,
то  $L=0$ есть стационарная точка передаточной  функции $T$.

Суперфункция $\mathcal F$ для передаточной функции $\mathcal T$ 
\be
F(z)=\mathcal F(z)-\mathcal L
\eL{FzFzL}
Схожим
образом преобразуется и
абельфункция  $G=F^{-1}$:
\be
G(z)=\mathcal G(z+\mathcal L)
\eL{GzGzL}
Читатель может проверить, что если в некоторой области значений $z$ выполняется
$F(G(z))\!=\!1$, то и в некоторой области значений $z$ выполняется
$\mathcal F(\mathcal G(z))\!=\!1$, и наоборот.
Таким образом, рассмотрения случая $T(L)\!=\!L$, $T^{\prime}(L)\!=\!1$ сводится к рассмотрению случая
 $T(0)\!=\!0$, $T^{\prime}(0)\!=\!1$, рассмотренного в предыдущей секции.
\vskip 2mm

Таким образом, угаданная асимптотика суперфункции позволяет быстро и аккуратно вычислять её, не напрягая особенно компьютер. Такой способ вычисления представляется порядков на 10 более эффективным, чем иные алгоритмы, которые удалось скачать из интернета.
Как это указано в Предисловии, такую неэффективность алгоритмов, рассмотренных в плоть до 2009 года, я считаю достаточным основанием и оправданием для того, чтобы не рассматривать и не описывать алгоритмы, предложенные коллегами, по крайней мере в этой книге. Если кто-либо их коллег предложит представление, которое ещё хотя бы на порядок величины эффективнее предложенного выше; то он, в свою очередь, может игнорировать представление этой секции как антикварное и не эффективное. 
\vskip 2mm 
}%JP

\JP{
В следующей главе, предложенные здесь формулы применяются для вычисления суперфункции экспоненты по основанию
$b\!=\! \exp(1/\mathrm e)$. %\approx 1.444667861009$.
Этот случай рассмотрен по просьбе Генрика Траппманна, и константу $\eta\!=\!\exp(1/\mathrm e)\!\approx\! 1.44466786101$ можно назвать константой Генрика или основанием Генрика 
\footnote
{ Название ``{\bf Основание Генрика}'' несколько двусмысленно, примерно как ``{\bf Защита Лужина}'', или ``{\bf Изобретение Вальса}'', или комбинация ``{\bf РОЙ МЕДВЕДЕВ --- Они кружились вокруг Сталина}''.
Вероятно, удачное название и должно быть двусмысленым, чтобы читатель подумал, как бы такое защитить Лужина, или о том, где и когда изобретён Вальс, или представил себе медведей (или даже ``медведёв''), которые, как пчёлы, роем кружат вокруг Сталина,
или попытался вспомнить,  кто и как основал Генрика. }. Карта такой экспоненты, карта её обратной функции и явный график для вещественных значений аргумента показаны в следующей главе на рисункax  11.2, 11.3, \ref{expe1eplot}. 
}%JP

\JP{
\begin{figure}[h]
\begin{center}
\sx{.5}{\begin{picture}(600,600)
\put(0,0){\ing{roy}}
\end{picture}}
\end{center}
\caption{РОЙ МЕДВЕДЕВ, кружатся вокруг Сталина \label{roy}}
\end{figure}
}

\section{Exponent to base $\exp(1/\mathrm e)$}

Let us
apply the formalism of exotic iterations above to the exponential to base
$b\!=\!\eta\!=\!\exp(1/\mathrm e)\!\approx\! 1.44466786~$.
This section deals with function $\mathcal T$ by
\be \!\!\!\!\!\!\!\!\!\!
\mathcal T(z)=\eta^z=\exp_{\eta} (z)=\exp(\ln(\eta)\, z)=\exp(z/ \mathrm e)
\eL{etaz} 
Explicit plot of this transfer function is shown in figure \ref{expe1eplot} with thick curve.
For comparison, the thin curve shows the exponent to base $\sqrt{2}$, considered above in chapter 9; this 
curve is borrowed from figure
\ref{ExpQ2plot}. 
The curves at this figure look close; at small and negative values of the argument, they almost overlap.
However, the deviation is significant at large positive values of the argument, and this determines the pretty different behaviour of the corresponding superfunctions.
\vskip 2mm

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\hskip 24pt
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\vskip -4pt
\caption{\large $u\!+\!\mathrm i v=\exp_{\eta}(x\!+\!\mathrm i y)~$
\iL{expe1emap}}
\end{figure}

Complex map of the transfer function  $\exp_{\eta}$  is shown in figure \ref{expe1emap}.
This map looks similar to the map at figure \ref{ExpQ2map}; the only period is slightly different.
For the base $\eta$, this period 
\be
P_{\exp_\eta}=2\, \pi \, \mathrm i \, \mathrm e \approx 17.0794684453 \, \mathrm i
\eL{Pexpeta}
It is amassing, that the simple formula \rf{Pexpeta} combines the 3 fundamental constants,
: $\pi$, $\mathrm i$ and $\mathrm e$. 
In addition, the Henryk base $\eta\!=\!\exp^2(-1)\!=\!\exp(1/\mathrm e)\!\approx\! 1.44466786101$ 
also is mathematical constant. A little bit less than one period $P_{\exp_\eta}$ fits the range 
of map in figure
\ref{expe1emap}.


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\iL{loge1emap}}
\vskip 26pt
\end{figure}
\vskip 6pt

For transfer function $\mathcal T\!=\!\exp_\eta$, the inverse function is 
$\mathcal T^{-1}\!=\! \exp_{\eta}^{-1}\!=\!\log_{\eta}$. 
Complex map of this function is shown in figure 
\ref{loge1emap}.
This map looks similar to the map of logarithm to base
$\sqrt{2}$, 
shown in chapter 9 at figure
\ref{LogQ2map}.
At that map, the mesh of isolines is a little bit more dense;
for example, all the line
$u\!=\!6$ 
happen to be in the range of the map,
while at the map at figure
\ref{loge1emap} 
the only part of this line is seen;
as for level
$v\!=\!\pm 9$ %для экспоненты по основанию Генрика вообще 
it happen to be beyond the cut line and, therefore, not seen in the map.

\vskip 2mm
Value $b\!=\!\eta$ is maximal real base, at which the exponent still has at least one real fixed point.
Below, for this exponent, the superfunction is constructed.
\section{Superexponent to base
$\eta\!=\!\mathrm e^{1/\mathrm e}\!=\!\exp^2(-1)$}
\baselineskip 20pt

This case had been considered by request from Henryk Trappmann.
He exopectes, that for the exponential to base 
\be
\eta=\exp(1/\mathrm e)=\exp^2(-1)\approx 1.44466786101
\eL{eta}
I cannot construct the efficient (fast and precise) algorithm of evaluation.
The similar opinion had been expressed in century 20 by Peter L. Walker \cite{walker}.
\vskip 6pt

In order to convince Henryk, we had to write the special article for the journal 
 %Чтобы переубедить Генрика, нам с ним пришлось писать специальную статью для журнала 
Mathematics of Computation \cite{e1e}.
Some formulas and figures from that publication are repeated
below.
\vskip 2mm

Let the transfer function $\mathcal T$ be defined with equation \ref{etaz} of the previous section; let
\be \!\!\!\!\!\! \!\!\!\!\!\!
\mathcal T(z)\!=\!\eta^z\!=\!\exp_{\eta} (z)\!=\!\exp\!\big(\ln(\eta)\, z\big)\!=\!\exp(z/ \mathrm e) 
\label{calTetaz}
\ee
In order to apply the formalism of exotic iteration from the previous section, I define the new, ``displaced''  transfer function,
as it is suggested by equation \rf{TnewTLzL};
for the initial transfer function $\mathcal T$ by
 \rf{calTetaz}.
To distinguish these transfer functions, I use slightly different fonts for the letter T in the names.
The new (displaced) transfer function
\be\!\!\!\!\!\!\!\!\!\!
T(z)\!=\!
\mathcal T
(z+\mathrm e)
\!-\!\mathrm e\!=\!\exp\big((z\!+\!\mathrm e)/\mathrm e\big)-\mathrm e=\exp(z/\mathrm e\!+\!1)\!-\!\mathrm e
\label{Te1e}
\ee
For this transfer function,
\be
T(0) &=&0    \\
T^{\prime}(0)&=& 1 \\
T^{\prime\prime}(0)&=& 2\,  v ~=~1/\mathrm e \\
T^{\prime\prime\prime}(0)&=& 6\, w ~=~ 1/\mathrm e ^2\\
 ... \\
T^{(n)}(0)&=&1/\mathrm e ^{n-1}
\ee
In the last formula, in the left hand side, the parenthesis in the superscript indicates not a number if iterate,
but number of derivative; function $T$ is differentiated $n$ times.

\vskip 2mm

For this case, the formalism of exotic iteration, described in the precious section, can be applied as is.
Expansion
\rf{TExpan} 
can be written as follows:
\be
T(z)=z+\frac{1}{2\mathrm e}z^2+\frac{1}{6 \mathrm e} z^3+..
\ee
For $\ds v\!=\!\frac{1}{2\mathrm e}$ and $\ds w\!=\!\frac{1}{6\mathrm e^2}$, formulas \rf{avbwv} give
\be
a&=&- 2\mathrm e\approx  - 5.43656365691809\\
b&=&-\frac{\mathrm e}{3} + \frac{1}{8\mathrm e^3} \approx   1.604598172578777
 %1.36787944117
\ee

Then, the primary approximation $f$ for superfunction \rf{FExpan} has the following form:
\be
\tilde f(z)&=& -\frac{1}{z} + \Big(1+\frac{1}{\mathrm e}\Big)\frac{\ln(\pm z)}{z^2}+..
\eL{fe1ez2}

More terms of this expansion can be calculated analytically, especially, if some Mathematica or Maple
are used.
In principle, the truncated series in \rf{fe1ez2}, even  with two terms, approximates the superfunction and can be used for its definition.
For $|z| \!>\! 100$, such a representation gives few correct decimal digits and can be used to plot the complex maps.
\vskip 2mm

For the accurate representation of superfunction, more terms in the expansion, similar to  \rf{fe1ez2}, should be calculated.
The asymptotic expansion can be written as follows:
\be \!\!\!\!\!\!\!\!
\tilde F(z)=\frac{-2\mathrm e}{z}\left(1+\sum_{m=1}^M \frac{P_m\big(\!-\ln(\pm z)\big)}{(3z)^m}+ \mathcal O\Big( \ln(\pm z)/z \Big)^{M+1} \right)
\label{e1e24}
\ee
where 
\be
P_m(x)=\sum_{n=0}^m c_{m,n} x^n
\eL{e1ePm}
One may choose the first coefficient $c_{0,0}\!=\!1$; then, $P_0(t)\!=\!1$.
Other coefficients $c$ are determined by the substitution of the expansion \rf{e1e24} into the transfer equation
\be
\tilde F(z\!+\!1)=\exp\Big(\tilde F(z)/\mathrm e+1\Big) - \mathrm e
\nonumber
\ee 
The first 5 polynomials $P$ are shown below:

\be
P_{1}(t)&=&t \\
P_{2}(t)&=&t^{2}+t+1/2 \\
P_{3}(t)&=&t^{3}+\frac{ 5}{ 2}t^{2}+\frac{  5}{2}t    +\frac{ 7}{10} \\
P_{4}(t)&=&t^{4}+\frac{13}{ 3}t^{3}+\frac{ 45}{6}t^{2}+\frac{53}{10}t    +\frac{ 67}{60} \\
P_{5}(t)&=&t^{5}+\frac{77}{12}t^{4}+\frac{101}{6}t^{3}+\frac{83}{ 4}t^{2}+\frac{653}{60}t+\frac{2701}{1680}
\label{Pc}
\ee

The superfunction $F$ appears as limit
\be
F(z)=\lim_{n\rightarrow \pm \infty} T^n (f(z-n))
\eL{e1elim}
where function $f$ is just truncation of the asymptotic expansion in  \rf{e1e24} at some positive integer $M$, and $n$ is chosen positive or negative,
dependently on the sign in the argument of logarithm in formula \rf{e1e24}.
\vskip 2mm

It is convenient to deal with functions, that have integer value at zero; so, I use the translation of the argument in order to define iteration 
$\mathrm{tet}_{\eta}$  and the growing sperexponential $\mathrm{SuExp}_{\eta,3}$ with the following formulas:
\be 
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{tet}_{\eta}(z)&=&F_1(z)=f(z\!+\!x_1)  \iL{e1etet}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{SuExp}_{\eta,3}(z)&=& F_3(z)=f(z\!+\!x_3)  \iL{e1eSuExp}
\ee
where constants
$x_1\!\approx\!  2.798248154231454$ and $x_3\!\approx\! -20.28740458994004$ 
are chosen to provide relations $F_1(0)\!=\!1$ and $F_3(0)\!=\!3$.
\vskip 2mm

Function  $\mathrm{tet}_{\eta}$ corresponds to the upper sign go the  $\pm$ in formuas \rf{e1e24}, \rf{e1elim},
while function $\mathrm{SuExp}_{\eta,3}$ by  \rf{e1eSuExp} refers to the lowerst sign.
In such a way,  $F_1$ and $F_3$ 
are pretty different functions, and it is difficult to express one of them through another.
\vskip 2mm

In formulas \rf{e1etet} and \rf{e1eSuExp}, for the two superfunctions $F_1$ and $F_3$, a little bit longer names 
$\mathrm{tet}_{\exp(1/\mathrm e)}$
and $\mathrm{SuExp}_{\exp(1/\mathrm e)}$
are suggested. These long names simplify the identification at the use from other chapters of this Book (and also from other publications), where the base $b$ may have various values, and not necessary
$b\!=\!\eta$.
\vskip 2mm

I repeat the meanings of the names suggested.
$\mathrm{tet}_{\eta}$, 
indicates, that this refers to tetration. For any tetration (to any base $b$),
I assume condition
$\tet_{b}(0)\!=\!1$. 
\vskip 2mm

The shoice of value at zero of the growing super exponential $F_3$ is not so obvious.
In publications
\cite{sqrt2,e1e} 
(and not only there), the value at zero is chosen as minimal integer, that is still larger, than the real fixed point.
Such a choice has sense, while the only one transfer function without parameters is considered, or if value of its parameter is fixed,
and no continuity properties with respect to this parameter are analysed.
However, if we treat base $b$ as parameter, then, the choice of integer value at zero leads to discontinuity with respect to base $b$.
For the case, if such a dependence will be requested, the notation
$\mathrm{SuExp}_{\eta,3}$; 
is used; the value at zero is indicated as the additional subscript.
\vskip 2mm

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\vskip -18pt
\hskip 9pt
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\vskip -2pt
\caption{\large 
Two super-exponentials to base 
$b\!=\!\exp(1/\mathrm e)$ 
\label{fig1}
\label{fige1e1}
~\iL{e1eplot}$\!\!\!\!$
}
\end{figure}

For the real argument, functions $F_3$ and $F_1$ are shown in figure \ref{e1eplot}.
Complex maps of these superfunctions are shown in figure  
\ref{e1emap}. 
These functions are pretty different.
In order to release the notation $F$ 
with subscripts for other functions, used in other chapters (and in other publications), I give them the special names
 $\mathrm{SuExp}_{\eta,3}\!=\!F_3$ and
$\mathrm{tet}_\eta\!=\!F_1$.
Symbol $\mathrm{tet}$ refers to tetration; in the following chapters, it is defined also for other values of base.
\vskip 2mm

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 \put(120,  40){\sx{1.1}{$v\!=\!-0.4$}}
 %\put(140,230){\sx{1.1}{$p\!=\!2.4$}}
 \put(214,110){\sx{1.1}{$u\!=\!2.4$}}
 %\put(140, 64){\sx{1.1}{$p\!=\!2.4$}}
 %\put(160,198){\sx{1.1}{$q\!=\!0.2$}}
 \put(190,260){\sx{1.1}{$v\!=\!0.2$}}
 \put(130,156){\sx{1.3}{$u\!=\!2$}}
 \put(277, 148.4){\sx{1.3}{$v\!=\!0$}}
 \put(158,132){\sx{1.1}{$u\!=\!2.2$}}
 \put(190, 17){\sx{1.1}{$v\!=\!-0.2$}}
 \end{picture}}
\end{center}
\vskip -4pt
\caption{\large \baselineskip 18pt
$u\!+\!\mathrm i v\!=\!\mathrm{SuExp}_{\eta,3}(\!x\!+\!\rmi y)\!=\!F_3(\!x\!+\!\rmi y)$ 
by
\rf{e1eSuExp}, at the top, and
$u\!+\!\mathrm i v\!=\!\mathrm{tet}_\eta(\!x\!+\!\rmi y)\!=\!F_1(\!x\!+\!\rmi y)$ 
by \rf{e1etet}, 
at the bottom.
\iL{e1efig2}
\label{figmapf}
\label{e1emap}
}
\end{figure}

Along the real axis, function $\mathrm{SuExp}_{\eta,3}$ 
infinitely grows at the positive values of the argument, and approaches its limiting value
$\mathrm e$ 
at large negative values. The function approaches to the same value in the most of the complex plane, 
except the strip along the positive direction od the real axis.
\vskip 2mm

Function $\mathrm{tet}_\eta$, at large values of the argument, approaches its limiting value  $\eta$ almost everywhere,
but has logarithmic singularity at $-2$, going to infinity at this point. Tetration to any base has this singularity;
it follows from the additional condition $\mathrm{tet}_b(0)\!=\!1$ and the transfer equation, that in vicinity of the real axis can be rewritten in the following form:
\be
\mathrm{tet}_b(z)\!=\!\log_b\!\Big(\mathrm{tet}_b(z\!+\!1)\Big)
\ee


\JP{{\bf Лирическое отступление.}
Когда я копипастил сюда рисунки, я понял, что присутствие в стране русскоязычных резидентов
может использоваться путинистами как предлог для вооруженного вторжения в страну, где я живу и работаю.
Поэтому те части, которые ещё не загружены, будут писаться по-английски. March 20.2014.
{\bf Warning.}
While I copy pasted these figures, I realise, that presence of the Russian-speaking residents in a country
can be used by putinists as a pretext for the military invasion. For this reason, since now (2014, March),
the Russian language is not recommended at all. Perhaps, the parts of this book, that ate not yet loaded, will be loaded in English.}%JP
\vskip 2mm

For iterates of exponent to base $\eta$, we need not only the superfunctions, but also the corresponding abelfunctions.
These abelfunstions are considered in the next section.

\section{Abelexponent to Hernyk base}
\baselineskip 18.2pt
Historically, the exponent to base $\eta=\exp(2/\mathrm e)$ is first function, for which the exotic iteration had been applied
\cite{e1e}. The two superfunctions for this exponent are specified in the previous section. Here, I describe the inverse functions,
id est, the two abelfunctions of the exponent to base $\eta$.
\vskip 2mm

Let $G_1=F_1^{-1}$ and $G_3=F_3^{-1}$. Maps of these functions are shown in figure 
\ref{e1efig3}.
{For evaluation of functions $G_1$ and $G_3$,  the asymptotic representations are used:}%EN
\begin{figure}
\begin{center}
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:E1egi4.jpg}
\vskip -1pt
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 \put(0,0){\ing{e1egi} }  % suppress below to boost
 \figaxe
 \put(  4,220){\sx{1.2}{$u\!=\!19$}} 
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 \put(352,276){\sx{1.2}{$v\!=\!0.2$}}
 \put(350, 26){\sx{1.2}{$v\!=\!-0.2$}}
 \put(350,186){\sx{1.2}{$u\!=\!18.4$}}
 \put(270,176){\sx{1.2}{$u\!=\!18.2$}}
 \put(233,158){\sx{1.2}{$u\!=\!18$}}
 \put(  4, 74){\sx{1.2}{$u\!=\!19$}} 
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 \put(20,148.4){\sx{1.2}{\bf cut }} 
 \put(377,148.4){\sx{1.1}{$v\!=\!0$}}
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 \vskip 10pt
\url{https://mizugadro.mydns.jp/t/index.php/File:E1eti4.jpg}
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 \put(0,0){\ing{e1eti} } % suppress to boost
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 \put(10,233){\sx{1.4}{$v\!=\!0$}} 
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 \put(142,225){\sx{1.001}{$u\!=\!-3$}}
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\end{center}
\vskip -4pt
\caption{\large \baselineskip 18pt
$u\!+\!\mathrm i v
\!=\!G_3(x\!+\!\mathrm i y)
\!=\! \mathrm{AuExp}_{\eta,3}(x\!+\!\mathrm i y)$ by \rf{e1eAuExp}, top map, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ and
$u\!+\!\mathrm i v
\!=\!G_1(x\!+\!\mathrm i y)
\!=\! \mathrm{ate}_\eta(x\!+\!\mathrm i y)$
by \rf{e1eate}, map at the bottom
\iL{e1efig3} ~ ~ ~ ~ ~
\label{e1efigi} $\!\!\!\!\!\!\!\!$
}
\end{figure}
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
g(z)=g_{\pm}(z)=
\frac{\ln(\pm t)}{3}+\frac{2}{t}+\sum_{n=1}^{15} c_n t^n + O(t^{16})
\eL{e1egpm}

{where}
 $t\!=\!(z-\rme)/\rme$.
{The coefficients $c$ can be found inverting the expansion for the superfunction, and also by
substituting expansion \rf{e1egpm} into the Abel equation}%EN
\be
g(z)+1=g\Big( \exp\big(g(z)/ \mathrm e\big) \Big)
\eL{e1eAbeleq}

{The truncation of the series, id est, omitting of $O$ in the right hand side of \rf{e1egpm}, provides 
the algorithm for evaluation of the abelfunction $g$ with at least 15 significant figures, while $|t|<1/2$.
For larger values, the argument of $g$ should be transformed, using the equation \rf{e1eAbeleq}.}%EN
\vskip 2mm

{The inverse functions $G_1\!=\!F_1^{-1}$ and $G_3\!=\!F_3^{-1}$ can be defined, adding the corresponding constants,}%EN
\be\!\!\!\!\!\!
\mathrm{ate}_{\eta}(z)
=G_1(z)=g(z)-g(1)\approx g(z) - 3.029297214418
\label{e1eate}
\label{e1eG1}
\ee
{where the upper sign in \rf{e1egpm} is used, and}%EN
\be\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{AuExp}_{\eta,3}\!=\!G_3(z)\!=\!g(z)\!-\!g(3)\!\approx\! g(z) \!+\! 20.0563555297533789
\label{e1eAuExp}
\ee
{where the lower sign in \rf{e1egpm} is used.}%EN
\vskip 2mm

{The additional names ate and AuExp are introduced in order to simplify identification and referencing to these functions from other chapters.}
\vskip 2mm

Representations \rf{e1eG1} and \rf{e1eAuExp} provide relations $G_1(1)\!=\!0$ and $G_3(3)\!=\!0$, as it is supposed to be for the inverse functions of $F_1$ and $F_3$.
The numerical tests confirm, that in wide ranges of values $z$,
the relations 
\be\!\!\!\!\!\!\!\!\!\!\!\!
F_1(G_1(z))\!=\!z\,  ,  ~
F_3(G_3(z))\!=\!z\,  ,  ~
F_1(G_1(z))\!=\!z\,  ,  ~
G_3(F_3(z))\!=\!z ~
\label{e1eId}
\ee
hold with 14 significant figures. The readers are invited to plot the map of the agreement function
\be\!\!
A(z)=-\lg\left( \frac{|F(G(z))-z|}{|F(G(z))|+|z|}\right)
\eL{e1eA}
{adding appropriate subscripts and exchanging}
$F\leftrightarrow G$,
{as numerical check of equalities in statement \rf{e1eId}. Levels of $A(x\!+\!\mathrm i y)$ in the $x,y$ plane indicate, where the algorithm works almost without loss of precision, id est, close to the maximal precision allowed for the complex double variables.}%EN
\vskip 2mm


\section{Iterates}
\baselineskip 18.2pt

{With function $F_3$ and $G_3$, one can express the iterates of exponential to base $\exp(1/\mathrm e)$. 
For $T(z)\!=\!\exp(z/\mathrm e)$,}
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!
T^n_{\mathrm u}(z)\!=\!\exp_{\eta,\mathrm u}^n(z)
\!=\!F_3(n\!+\!G_3(z))
\!=\!\mathrm{SuExp}_{\eta,3}\Big(n\!+\!\mathrm{AuExp}_{\eta,3}(z)\Big)
\label{e1eF3G3}
\ee
Here, the subscript $u$ indicates, that the this iterate is holomorphic at the highest, upper range of the real values of the argument, and, in particular, in the vicinity of the half-line $z\!>\!\mathrm e$.
\vskip 2mm

{In the representation \rf{e1eF3G3}, neither
argument $z$ nor number of iterate has need to be real. 
For $n\!=\!1/2$, the complex map of the half iteration of exponential to base $\exp(1/\mathrm e)$ is shown in figure \rf{e1eghalf}.}%EN
\vskip 2mm

{For real values of argument $T^n_{\mathrm u}(x)$ versus $x$ for various values of number $n$ of iteration is shown in figure \ref{e1eiter}. The thick curves correspond to the integer values of $n$. The non-integer iterates are holomorphic at least in some vicinity of half-line  $x\!>\!\mathrm e$. 
The integer iterates can be extended to $-\infty$ for positive $b$ or to the closest vertical asymptotic of the corresponding logarithm for negative number of iteration. The 
bisector 
of the First quadrant of the coordinate plane corresponds to the identity function.}%EN

\vskip 2mm
The non integer iterates by \rf{e1eF3G3} are shown in figure \ref{e1eiter} with thin lines.
These lines cannot be extended beyond the fixed point $\mathrm e$; this is branch point of the non-integer iterates.
These lines remain in the range
$x\!>\!\eta$, $y\!>\!\eta$; 
in particular, they cannot be holomorphically extended into the third quadrant.
\vskip 2mm

\begin{figure}[t]
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\vskip 4pt
\hskip 36pt
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\vskip -4pt
\caption{\large
$y=\exp_{\eta,\mathrm u}^{~n}(x)$ 
{by \rf{e1eF3G3} versus $x$ for various $n$}%EN
\iL{e1eiter} }
\end{figure}
\vskip 2mm

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\vskip 6pt
\url{https://mizugadro.mydns.jp/t/index.php/File:E1eghalfm3.jpg}

\caption{$u\!+\!\mathrm i v=\exp_{\eta,\mathrm u}^{1/2}(x\!+\!\mathrm i y)$ 
by
\rf{e1eF3G3} 
for
$n\!=\!1/2$  \iL{e1eghalf}}
\end{figure}

The non-integer iterates by formula \rf{e1eF3G3}
have cut line along the real axis from $-\infty$ to $\mathrm e\approx 2.71$;
in order to shown this cut, figure \ref{e1eghalf} represent the complex map of the half iterate, $n\!=\!1/2$.
This cut is traced with dashed line and 
labeled
with symbol
{\bf cut}.
\vskip 2mm

As other complex maps of other iterates of a real-holomorphic function, the map at figure 
\ref{e1eghalf} 
is symmetric with respect to reflection from the real axis, 
\be
\exp_{\eta,\mathrm u}^{1/2}(z^*)=
\exp_{\eta,\mathrm u}^{1/2}(z)^*
\ee
In order to remind this symmetry, I plot the maps, placing the real axis at the centre of the figure.
However, sometimes, there is not enough space at the page, then I plot the only upper part of the map.
I hope, the Reader has enough imagination, that allows him or her to consider 
the imaginary mirror, and see in this imaginary mirror the function with inverted sign of the its imaginary part.
With hopes for such imagination, in Figure \ref{e1e000map}, I show the iterates of the exponent to base $\eta$ for various number of iterate $n$
in a similar way, as the case $n\!=\!1/2$ is shown in Figure \ref{e1eghalf}. However, the only halts of the maps are shown in Figure
\ref{e1e000map}.
\vskip 2mm

\section{Not only exp}

Exponential to base $\eta=\exp(1/\mathrm e)=\exp^2(-1)$ is not only transfer function, that have derivative unity at its fixed point, $T'(L)=1$.
Some of such transfer functions can be treated with the method of exotic iteration, described in this chapter.
\vskip 8pt

One more example is considered in the next chapter, referring to the elementary transfer function $T(x)=\mathrm{zex}(x)=z \exp(z)$.
It is treated in a pretty similar way, as $\exp_\eta$.
\vskip 8pt

Then, in the following chapters, even more exotic cases are treated, when $T'(L)=1$ and $T''(L)=0$ also can be treated.
However, I try to go step by step.
So, open the next chapter and read about iterates of zex.

\vskip 2mm

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\put(146,112){\rot{41} \sx{5}{$u\!=\!2$} \ero}
\put(204, 62){\rot{41} \sx{5}{$u\!=\!1$} \ero}
\put(258, 35){\rot{38} \sx{5}{$u\!=\!0$} \ero}
\put(542,236){\rot{57} \sx{5}{$v\!=\!3$} \ero} 
\put(606,166){\rot{39} \sx{5}{$v\!=\!2$} \ero} 
\put(644,80){\rot{19} \sx{5}{$v\!=\!1$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}}
\vskip 8pt

~
\sx{.25}{\begin{picture}(802,402) 
\put(0,0){\ing{e1e008map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.8$}}
\put( 46,152){\rot{35} \sx{5}{$u\!=\!0$} \ero}
\put(194,100){\rot{53} \sx{5}{$u\!=\!0.2$} \ero}
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\put(444,060){\rot{65} \sx{5}{$u\!=\!1$} \ero}
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\put(682,314){\rot{-46} \sx{5}{$v\!=\!3$} \ero}
\put(680,192){\rot{-30} \sx{5}{$v\!=\!2$} \ero}
\put(678,86){\rot{-15} \sx{5}{$v\!=\!1$} \ero}
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}} ~
\sx{.25}{\begin{picture}(802,402) 
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\put(20,340){\sx{7}{$n\!=\!-0.8$}} 
\put(92,192){\rot{29} \sx{5}{$u\!=\!3$} \ero}
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\put(8,-13){\sx{5}{\bf cut}} 
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}}
\vskip 8pt

~
\sx{.25}{\begin{picture}(802,402) \put(0,0){\ing{e1e005map}} \LeLa 
\put(20,340){\sx{7}{$n\!=\!0.5$}} 
\put(278,066){\rot{67} \sx{5}{$u\!=\!0$} \ero}
\put(482,054){\rot{76} \sx{5}{$u\!=\!1$} \ero}
\put(616,048){\rot{80} \sx{5}{$u\!=\!2$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\put(686,388){\rot{-34} \sx{5}{$v\!=\!4$} \ero}
\put(680,288){\rot{-26} \sx{5}{$v\!=\!3$} \ero}
\put(680,185){\rot{-16} \sx{5}{$v\!=\!2$} \ero}
\put(680,86){\rot{-9} \sx{5}{$v\!=\!1$} \ero}
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}} ~
\sx{.25}{\begin{picture}(802,402) \put(0,0){\ing{e1em05map}}  
\put(20,340){\sx{7}{$n\!=\!-0.5$}}  
\put(412,372){\rot{-45} \sx{5}{$u\!=\!2$} \ero}
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\put(598,280){\rot{29} \sx{5}{$v\!=\!3$} \ero}
\put(638,186){\rot{18} \sx{5}{$v\!=\!2$} \ero}
\put(666, 86){\rot{10} \sx{5}{$v\!=\!1$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}}
\vskip 8pt

~
\sx{.25}{\begin{picture}(802,402) \put(0,0){\ing{e1e002map}} 
\put(20,340){\sx{7}{$n\!=\!0.2$}}  \LeLa %\DoLa
\put(74,158){\rot{56} \sx{5}{$u\!=\!-2$} \ero}
\put(214,082){\rot{66} \sx{5}{$u\!=\!-1$} \ero}
\put(378,066){\rot{79} \sx{5}{$u\!=\!0$} \ero}
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\put(616,048){\rot{84} \sx{5}{$u\!=\!2$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\put(682,376){\rot{-12} \sx{5}{$v\!=\!4$} \ero}
\put(680,282){\rot{-10} \sx{5}{$v\!=\!3$} \ero}
\put(680,185){\rot{-7} \sx{5}{$v\!=\!2$} \ero}
\put(680,86){\rot{-4} \sx{5}{$v\!=\!1$} \ero}
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}} ~
\sx{.25}{\begin{picture}(802,402) \put(0,0){\ing{e1em02map}}  
\put(20,340){\sx{7}{$n\!=\!-0.2$}}  %\DoLa 
\put(162,282){\rot{-62} \sx{5}{$u\!=\!-2$} \ero}
\put(248,312){\rot{-67} \sx{5}{$u\!=\!-1$} \ero}
\put(344,318){\rot{-69} \sx{5}{$u\!=\!0$} \ero}
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\put(8,-13){\sx{5}{\bf cut}} 
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}}\vskip 8pt

~
\sx{.25}{\begin{picture}(802,402) \put(0,0){\ing{e1e001map}}  \put(20,340){\sx{7}{$n\!=\!0.1$}}  \LeLa \DoLa
\put(290,150){\rot{78} \sx{5}{$u\!=\!-1$} \ero}
\put(402,136){\rot{79} \sx{5}{$u\!=\!0$} \ero}
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\put(678,285){\rot{-6} \sx{5}{$v\!=\!3$} \ero}
\put(674,186){\rot{-4} \sx{5}{$v\!=\!2$} \ero}
\put(672,86){\rot{-1} \sx{5}{$v\!=\!1$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\put(682,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\end{picture}} ~
\sx{.25}{\begin{picture}(802,402) \put(0,0){\ing{e1em01map}}  
\put(20,340){\sx{7}{$n\!=\!-0.1$}}  \DoLa 
\put(176,310){\rot{-76} \sx{5}{$u\!=\!-2$} \ero}
\put(269,326){\rot{-78} \sx{5}{$u\!=\!-1$} \ero}
\put(366,350){\rot{-81} \sx{5}{$u\!=\!0$} \ero}
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\put(562,386){\rot{-84} \sx{5}{$u\!=\!2$} \ero}
\put(654,289){\rot{6} \sx{5}{$v\!=\!3$} \ero}
\put(662,185){\rot{5} \sx{5}{$v\!=\!2$} \ero}
\put(670,87){\rot{1} \sx{5}{$v\!=\!1$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\caption{\large
$u\!+\!\mathrm i v=\exp_{\eta,\mathrm u}^{~n}(x\!+\!\mathrm i y)$, complex maps of iterates of exponent to base $\eta$
by equation
\rf{e1eF3G3}
for various
$n$
\iL{e1e000map}}
\end{figure}

\chapter{~ LambertW and zex}

\baselineskip 20pt

\begin{wrapfigure}[1]{r}{408px}
\hskip -20pt
\normalsize
\sx{.64}{\begin{picture}(608,380)
 \put(1,9){\ing{zexplo2}}
 \put(388,398){\sx{2.9}{$y$}}
 \put(409,378){\sx{3.1}{$\mathrm e$}}
 \put(388,303){\sx{2.8}{$2$}}
 \put(388,203){\sx{2.8}{$1$}}
 \put(388,103){\sx{2.8}{$0$}}
 \put(409,70){\sx{2.6}{$-\!1/\mathrm e$}}
 \put( 86, 90){\sx{2.5}{$-\!3$}}
 \put(186, 90){\sx{2.5}{$-\!2$}}
 \put(286,90){\sx{2.5}{$-\!1$}}
 % \put(300,-9){\sx{2.5}{$0$}}
 \put(500,90){\sx{2.5}{$1$}}
 \put(600,90){\sx{2.5}{$2$}}
 \put(630,90){\sx{2.6}{$x$}}
 \put(455,180){\sx{3}{$y\!=\!\mathrm{zex}(x)$}}
 \end{picture}}
 \vskip -12pt
\hskip 10pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ZexPlot.png}
\vskip -4pt 
\caption{\large
$~y\!=\!\mathrm{zex}(x)\!=\!x \, \exp(x)$ \iL{zexplot}}
\vskip -300pt
\end{wrapfigure}
~

\rule{0pt}{310pt}

This chapter deals with function
ArcLambertW,
\be 
\!\!\!\!\!\!\!\!
\mathrm{ArcLambertW}(z) = \mathrm{ArcProductLog}(z) 
= \mathrm{zex}(z) = z \exp(z)
\label{zexDef}
\ee
\begin{wrapfigure}{r}{60pt}
\normalsize
\sx{.3}{\begin{picture}(200,90)
\put(0,0){\ing{monoxid}}
\put(84,64){\sx{4}{O}}
\put(24,30){\sx{4}{H}}
\put(144,30){\sx{4}{H}}
\end{picture}}
\vskip -8pt
\caption{\baselineskip 13pt Dihidrogen monoxide \label{monoxid}}
\vskip -12pt
\end{wrapfigure}
shown in figure \ref{zexplot}. 
The long name 
ArcLambertW 
is equivalent to even longer name
ArcProductLog. 
This name looks so  long, as ``Logistic map'' or ``dihidrogen monoxide''
shown in figure \ref{monoxid}. 
Formally, the name is correct; the inverse function of ProductLog can be called ArcProductLog in analogy with arcsin
or ArcTetration. 
But name zex 
is shorter.\\
Complex map of function zex is shown in figure \ref{zexmap}.
\vskip 2mm

Short name zex is created from the first three characters in the right hand side of equation \rf{zexDef}. Hope, this long explanation helps to avoid confusions. In the chapter name, I keep the indication to LambertW, because it is used in many publications and algorithmic languages.
As the negative iterates are allowed, if one can iterate zex, one can iterate also LambertW. Function zex is simpler than LambertW, so, I consider zex first.
\vskip 2mm

\begin{figure}[t]
\sx{.34}{\begin{picture}(1252,1220)
\put(40,20){\ing{zexmap}}
\put(18,1206){\sx{3.3}{$y$}}
\put(18,1113){\sx{3}{$5$}}
\put(18,1013){\sx{3}{$4$}}
\put(18, 913){\sx{3}{$3$}}
\put(18, 813){\sx{3}{$2$}}
\put(18, 713){\sx{3}{$1$}}
\put(18, 613){\sx{3}{$0$}}
\put(-6, 513){\sx{3}{$-1$}}
\put(-6, 413){\sx{3}{$-2$}}
\put(-6, 313){\sx{3}{$-3$}}
\put(-6, 213){\sx{3}{$-4$}}
\put(-6, 113){\sx{3}{$-5$}}
\put(-6, 013){\sx{3}{$-6$}}
\put(014, -5){\sx{3}{$-6$}}
\put(114, -5){\sx{3}{$-5$}}
\put(214, -5){\sx{3}{$-4$}}
\put(314, -5){\sx{3}{$-3$}}
\put(414, -5){\sx{3}{$-2$}}
\put(514, -5){\sx{3}{$-1$}}
\put(635, -5){\sx{3}{$0$}}
\put(735, -5){\sx{3}{$1$}}
\put(835, -5){\sx{3}{$2$}}
\put(935, -5){\sx{3}{$3$}}
\put(1035, -5){\sx{3}{$4$}}
\put(1135, -5){\sx{3}{$5$}}
\put(1227,-4){\sx{3}{$x$}}
\put(90,1162){\sx{4}{\rot{7}$u\!=\!0$\ero}}
\put(90,987){\sx{4}{\rot{6}$v\!=\!0$\ero}}
\put(90,802){\sx{4}{\rot{5}$u\!=\!0$\ero}}
\put(90,611){\sx{4}{$v\!=\!0$}}
\put(404,545){\sx{4}{\rot{90}$u\!=\!-0.2$\ero}}
\put(980,641){\sx{4}{\rot{-4}$v\!=\!16$\ero}}
\put(970,581){\sx{4}{\rot{3}$v\!=\!-16$\ero}}
\put(978,517){\sx{4}{\rot{-7}$u\!=\!16$\ero}}
\put(970,458){\sx{4}{\rot{-1}$u\!=\!-16$\ero}}
\put(90,421){\sx{4}{\rot{-5}$u\!=\!0$\ero}}
\put(90,237){\sx{4}{\rot{-6}$v\!=\!0$\ero}}
\put(90, 61){\sx{4}{\rot{-7}$u\!=\!0$\ero}}
\end{picture}}
\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ZexD6mapT100.png}
\vskip -6pt
\caption{$u\!+\!\mathrm i v \!=\! \mathrm{zex}(x\!+\!\mathrm i y)$ by \rf{zexDef} \iL{zexmap}}
\vskip 10pt
\end{figure}

Consideration in this chapter is very similar to that of the previous one; the similar exotic iterates are constructed.
I hope, the reader can perform the calculus for iterates of zex in the same way, as the deduction is performed in the previous chapter for
the exponent to base $\eta\!=\!\mathrm e^{1/\mathrm e}$. 
On the other hand, I remember the old rule: 

\baselineskip 12pt
If at your presentation, at the first desk you see Hideki Yukawa and Sofia Kovalevskaya, and at the last desk you see some Bart Simpson and Shoko Okudaira, then you should address to Bart and Shoko. In this case, you may hope, that Hideki and Sofia understand at least the main idea of your talk. 
\baselineskip 19pt
\vskip 6pt

Adopting that rule to the Book, here I repeat the deduction of the previous section for function zex. 
However, I would like the Readers at least to try to make this deduction by themselves, using the Book only to verify the results; then the Readers will be able to do the same also for other functions (including exotic ones).

\section{Holomorphic zex}
\baselineskip 20.8pt
For real argument, function zex by \rf{zexDef} is shown in figure \ref{zexplot}.
At negative values of the argument,  zex is negative, and it is positive for positive argument. 
At large negative values of the argument,  zex decreases, reaches its local minimum
$-1/\mathrm e$ 
at $-1$, then grows up.
The graphic passes through point (0,0); zero is fixed point. At zero, its derivative is unity; so
the iterates of zex are qualified as exotic; the non-integer iterates are not regular at the fixed point.
Here, it may be a good moment to go a little bit back, to the regular iteration, in order to remember, what happens, for example, at the right hand side of equation \rf{a2.39}, when the derivative $T'$ of the transfer function at the fixed point becomes zero.

\vskip 2mm


\begin{figure}[t]
 %\sx{1.44}{ \begin{picture}(164,165)
 \sx{2.44}{ \begin{picture}(164,165)
 %\put(6,5){\ing{taniacontour}}
 \put(6,5){\ing{lambertwmap}}
 \put(2,162){\sx{.7}{$y$}}
 \put(2,144){\sx{.6}{$6$}}
 \put(2,124){\sx{.6}{$4$}}
 \put(2,104){\sx{.6}{$2$}}
 \put(4,118){ \sx{.8}{\rot{-36}$v\!=\!1.8$\ero}}
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 \put(8, 84){\sx{.8}{\bf cut}}
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 \put(-3,44){\sx{.6}{$-4$}}
 \put(-3,24){\sx{.6}{$-6$}}
 \put( 22,0){\sx{.6}{$-6$}}
 \put( 42,0){\sx{.6}{$-4$}}
 \put( 62,0){\sx{.6}{$-2$}}
 \put( 86,0){\sx{.6}{$0$}}
 \put(106,0){\sx{.6}{$2$}}
 \put(126,0){\sx{.6}{$4$}}
 \put(146,0){\sx{.6}{$6$}}
 \put(164,0){\sx{.7}{$x$}}
 \put( 43, 73){\rot{-39}\sx{.8}{$u\!=\!0.8$}\ero}
 \put( 41, 57){\rot{-27}\sx{.84}{$u\!=\!1$}\ero}
 \put( 39, 40){\rot{-19}\sx{.8}{$u\!=\!1.2$}\ero}
 \put( 33, 21){\rot{-18}\sx{.8}{$u\!=\!1.4$}\ero}
 \put( 85, 145){\rot{83}\sx{.82}{$v\!=\!1$}\ero}
 \put(137, 102){\rot{17}\sx{.8}{$v\!=\!0.2$}\ero}
 \put(144, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}
 \put(140, 65){\rot{-16}\sx{.72}{$v\!=\!-\!0.2$}\ero}
 \put(134, 46){\rot{-32}\sx{.72}{$v\!=\!-0.4$}\ero}
 \put( 79, 33){\rot{-80}\sx{.86}{$v\!=\!-1$}\ero}
 \end{picture}}
\vskip 2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:LambertWmap150.png}
\vskip -2pt
\caption{ \large
$u\!+\!\mathrm i v \!=\! \mathrm{ArcZex}(x\!+\!\mathrm i y)\!=\! \mathrm{LambertW}(x\!+\!\mathrm i y)$
\label{arczexmap}}
\vskip 12pt
\end{figure}

Complex map of function zex is shown in figure
\ref{zexmap}.
This function is entire, it has no singularities.
Map of the inverse function
\be
\mathrm{zex}^{-1}=\mathrm{ArcZex}=\mathrm{zex}^{-1}=\mathrm{LambertW}
\ee
is shown in figure \ref{arczexmap}. This function has cut, along the negative part of the real axis. This cut is marched with dashed line. 
\vskip 2mm

Function LambertW can be defined as solution $F$ of the differential equation
\be
F'(z)= \frac{F(z)}{(1+F(z))~z}
\eL{LambertWdifur}
with additional condition
$F(0)\!=\!0$,
where the contour of integration goes along the imaginary axis from zero to the imaginary part of argument $z$, and then, parallel to the real axis, to the value $z$. 
Solving the equation \rf{LambertWdifur},
the Reader may verify, that the inverse function of the solution is zex.
Note, that the similar contour of integration is used in chapter 5 to define function Tania by equation \rf{taniaprim}.
Indeed, function LambertW is related to the Tania function:
\rf{taniaprim}:
\be \!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{LambertW}(z)=\mathrm{Tania}\Big(\ln(z)-1\Big)
\eL{taniaLambertW}
As the imaginary part of the logarithm is limited to the $-\pi,\pi$ range, in this expression, the argument of the Tania function is always inside the strip along the real part, seen in the complex map of the Tania function in figure
\rf{taniaLambertW}. 
Function LambertW is in some sense simpler, than function Tania: it has only one cut line (and one branch point $-1/\mathrm e$), while the Tania functions has two.
\vskip 2mm

I could not find good implementation complex double for function LambertW in C++ (in which I have good plotters of complex maps). 
For this reason, I use the expansions listed below.
\vskip 2mm

For small values of argument, LambertW can be expanded as follows:

\be \!\!\!\!\!\!\!\!\!\!\!\! 
\mathrm{LambertW}(z)\!\!&\!=&
z\sum_{n=0}^{\infty} \frac{(n\!+\!1)^{n-1}}{n!} (-z)^n 
\nonumber\\
&\!=&\!\!
z \!-\! z^2 \!+\!\frac{3 z^3}{2}
\!-\!\frac{8z^4}{3}
\!+\!\frac{125z^5}{24}
\!-\!\frac{54 z^6}{5}
\!+\!\frac{16807 z^7}{720} +.. ~ ~
\ee
The series converges at
$|z|\!<\! 1/\mathrm{e} \!\approx\! 0.367879~$.
With 48 terms, at $|z|\!\le 0.2$, the truncated sum provides at least 16 correct decimal digits.
\vskip 2mm

The expansion at the branch point can be written as follows:
\be \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! 
&\ds \mathrm{LambertW}\Big(\frac{-1}{\mathrm e} + \frac{t^2}{2\mathrm e}\Big)
\!=-1
+t 
-\frac{t^2}{3} 
+\frac{11 t^3}{72} 
-\frac{43 t^4}{540} 
+\frac{769 t^5}{17280} 
- \frac{221 t^6}{8505}  \!\!\!\!  \nonumber\\ 
\!\!\!\!\!& \ds \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\! 
+\frac{680863 t^7}{43545600} 
-\frac{1963 t^8}{204120} 
+\frac{226287557 t^9}{37623398400} 
-\frac{5776369 t^{10}}{1515591000} +.. \!\!\!\!\!\!\!\!\!\!\!\!
\ee
The expansion can be used for approximation of LambertW while
$|t|\!<\!1$, 
{id est, while the argument of LambertW is close to}
 $-\exp(-1)\!\approx\!  -0.367879~$. 
\vskip 2mm

For large values of $|z|$, using notations $L\!=\ln(z)$ and $M\!=\!\ln^2(z)$  the expansion of $\mathrm{LambertW}(z)$ can be written as follows:

\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{LambertW}(z)~ = ~ ~ ~ ~ ~ L - M +\frac{M}{L}
+ ~ \frac{M(-2+M)}{2L^2} ~ ~ ~ ~\nonumber\\
+\, \frac{M(6-9M+2M^2)}{6L^3} ~~~~\nonumber\\
+\, \frac{M(-12+36M-22M^2+3M^3)}{12 L^4} ~~~~\nonumber\\
+\, \frac{M(60-300M+350M^2-125M^3+12M^4)}{60L^5}~~~~\nonumber\\
+\frac{M(-120+900M-1700M^2+1125M^3-274M^4+20 M^5)}{120L^6} ~~~~\nonumber\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+~ O\Big(\frac{M}{L}\Big)^7 ~ ~ ~
\label{LambertWLM}
\ee
where the effective parameter of expansion happens to be
$\varepsilon\!=\!\ln^2(z)/\ln(z)$; 
at $|\varepsilon| \!\ll\! 1$, the asymptotics \rf{LambertWLM} can be used for the evaluation of LambertW.
Here, as usually, the upper superscript after the function indicates the number of iterations, and the upper superscript after the argument and the closing parenthesis indicates the power that is assumed to be evaluated after the evaluation of the function.
However, $\ln^{-1}(z)=\exp(z)$ should not be confused with $1/\ln(z)$, nor
$\ln^{2}(z)=\ln(\ln(z))$ should be confused with $\ln(z)^2$, and so on.
\vskip 2mm

The above asymptotics leads to the efficient (fast and precise) algorithm for evaluation of LambertW. The C++ implementation is loaded as
\url{https://mizugadro.mydns.jp/t/index.php/LambertW.cin}.
\vskip 2mm


With functions zex and $\mathrm{LambertW}\!=\!\mathrm{zex}^{-1}$, the superfunction of zex and its abelfunction can be implemented.
The superfunction is considered in the next section.

\section{SuZex}

{Superfunction for the transfer function zex is solution $F$ of the transfer equation}
\be
F(z\!+\!1)=\mathrm{zex}\Big(F(z)\Big)
\eL{suzexFeq}
{For some integer $M>1$, search for the asymptotic expansion of the solution in the following form:}%
\be \!\!\!\!\!\!\!\!\!\!
F(z)=F_M(z)+ O\left(\frac{\ln(\pm z)^{M+1}}{z^{M+2}}\right) ~~~~~~~
\eL{SuZexFMa}
{where}%
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!
F_M(z)=  -\frac{1}{z} + \frac{\frac{1}{2}\ln(\pm z) }{ z^2} +
\frac{1}{z}
\sum_{m=2}^{M} \frac{P_m\Big(\ln(\pm z)\Big)}{z^m} ~
\eL{SuZexFM}

\be 
P_m(z)=\sum_{n=0}^{m} c_{m,n} (-z)^n ~~~~~~~~~~ ~~~~~~~~~~~
\eL{SuZexP}

{The coefficients $c$ can be found by substitution of expansion \rf{SuZexFMa} into the transfer equation \rf{suzexFeq}. 
This can be done with the Mathematica code below:

\vskip -12pt ~
\baselineskip 20pt
\begin{verbatim}
zex[z_] = z Exp[z];
Foo[z_] = -1/z + a Log[z]/z^2
Soo = Series[Foo[z+1]-zex[Foo[z]], {z,Infinity,3}]
Eoo = Coefficient[Soo,1/z^3]
Ao = Extract[Solve[Eoo==0, a], 1]
F2o[z_] = ReplaceAll[Foo[z], Ao]
F20[z_] = F2o[z] + (a Log[z]^2 + b Log[z] + c)/z^3
S2o = Series[F20[z+1] - zex[F20[z]], {z,Infinity,4}]
S20 = ReplaceAll[S2o, Log[1/z] -> -L]
E2o = Coefficient[S20, 1/z^4]
E22 = Coefficient[E2o, L^2]
A1 = Extract[Extract[Solve[E22==0, a], 1], 1]
E2A = ReplaceAll[E2o, A1]
E21 = Coefficient[E2A, L]
B1 = Extract[Extract[Solve[E21==0, b], 1], 1]
E2B = ReplaceAll[E2A, B1]
C1 = Extract[Extract[Solve[E2B==0, c], 1], 1]
F3o[z_] = ReplaceAll[F20[z], {A1, B1, C1}]
F30[z_] = F3o[z]+(a Log[z]^3+b Log[z]^2+c Log[z]+d)/z^4
S3o = Series[F30[z+1] - zex[F30[z]], {z, Infinity, 5}]
S30 = ReplaceAll[S3o, Log[1/z] -> -L]
E3o = Coefficient[S30, 1/z^5]
E33 = Coefficient[E3o, L^3]
A3 = Extract[Extract[Solve[E33==0, a], 1], 1]
E3a = ReplaceAll[E3o, A3]
E32 = Coefficient[E3a, L^2]
B3 = Extract[Extract[Solve[E32==0, b], 1], 1]
E3b = ReplaceAll[E3a, B3]
E31 = Coefficient[E3b, L]
C3 = Extract[Extract[Solve[E31==0, c], 1], 1]
E3c = ReplaceAll[E3b, C3]
D3 = Extract[Extract[Solve[E3c == 0, d], 1], 1]
F4o[z_] = ReplaceAll[F30[z], {A3, B3, C3, D3}]
F40[z_] = F4o[z] +
      (a Log[z]^4+b Log[z]^3+c Log[z]^2+d Log[z]+e)/z^5
S4o = Series[F40[z+1] - zex[F40[z]], {z, Infinity, 6}]
\end{verbatim}
\baselineskip 19pt
{Such a calculus leads to the following asymptotics:}%
\be \!\!\!\!\!\!
F(z)\!&\!=\!&  -\frac{1}{z}
+
\frac{\frac{1}{2}\ell }{ z^2}
+
\frac{\frac{-1}{4}\ell^2 +\frac{1}{4}\ell-\frac{1}{6}}{z^3_{\phantom{p}} } 
+ 
\frac{\frac{ 1}{8}\ell^3 +\frac{-5}{16}\ell^2+\frac{3}{8}\ell+\frac{-7}{48}  }{z^4_{\phantom{p}}} 
\nonumber \\ &&
+ ~
\frac{\frac{-1}{16}\ell^4+\frac{13}{48}\ell^3+\frac{-17}{32}\ell^2+\frac{23}{48}\ell+\frac{-707}{4320} }{z^5_{\phantom{p}}}
\nonumber \\ &&
+~
\frac{\frac{ 1}{32}\ell^5+\frac{-77}{384}\ell^4+\frac{37}{64}\ell^3+\frac{-83}{96}\ell^2+\frac{1121}{1728}\ell + \frac{-1637}{8640}  }{z^6_{\phantom{p}}} 
\nonumber \\ &&
+ ~
\frac{\frac{-1}{64}\ell^6+\frac{87}{640}\ell^5+\frac{-205}{384}\ell^4+\frac{443}{384}\ell^3+\frac{-1619}{1152}\ell^2+\frac{15427}{17280}\ell + \frac{-274133}{1209600}  }{z^6} \nonumber \\ &&
+ ~
O\left(\frac{\ell^7}{z^8}\right)
\ee
where $\ell=\ln(-z)$.
\vskip 2mm

\begin{wrapfigure}{r}{183px}
\normalsize
\vskip -32pt
\sx{.33}{\begin{picture}(1202,1102)
\put(322,1090){\sx{4}{$y$}}
\put(292,372){\sx{4}{$\mathrm e$}}
\put(315, 62){\sx{4}{$-\!1/\mathrm e$}}
\put(325,998){\sx{4}{$9$}}
\put(325,898){\sx{4}{$8$}}
\put(325,798){\sx{4}{$7$}}
\put(325,698){\sx{4}{$6$}}
\put(325,598){\sx{4}{$5$}}
\put(325,498){\sx{4}{$4$}}
\put(325,398){\sx{4}{$3$}}
\put(325,298){\sx{4}{$2$}}
\put(325,198){\sx{4}{$1$}}
\put(078,116){\sx{4}{$-\!2$}}
\put(178,116){\sx{4}{$-\!1$}}
\put(304,116){\sx{4}{$0$}}
\put(404,116){\sx{4}{$1$}}
\put(490,116){\sx{4}{$x$}}
\put(12,188){\sx{4}{\rot{0}$y\!=\!\mathrm{SuZex}(x)$\ero}}
\put(358,162){\sx{4}{\rot{0}$y\!=\!x\mathrm{e}^x$\ero}}
\put(10,10){\ing{suzexplo}}
\put(555,8){\rot{90}\sx{2.8}{\url{https://mizugadro.mydns.jp/t/index.php/File:SuZexPlot511T.jpg}}\ero}
\end{picture}}
\vskip -6pt
\caption{\large $~ ~y=\mathrm{SuZex}(x)~ ~  ~ ~ ~ ~ $ 
\label{suzexplo} }
\vskip 24pt
\end{wrapfigure}

For some number $M$ of terms taken into account, the superfunction $F$ can be expressed through its $M$th asymptotic as follows:
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F(z) =\lim_{n\rightarrow \infty} \mathrm{zex}^n\!\Big(F_M(z\!-\!n)\Big)
\label{SuZexFdef}
\ee
The resulting $F$ does not depend on the number $M$ of terms taken into account in the primary approximation. However, the convergence of the limit for larger $M$ is faster.
\vskip 2mm

In order to simplify the comparison of different representations of the superfunction, it is convenient to  
define the misplaced function
\be
\mathrm{SuZex}(z)=F(z_1\!+\!z)
\label{SuZexDef}
\ee
where $z_1\!\approx\! -1.1259817765745026$ is solution of equation $F(z_1)\!=\!1$.
This definition gives a way to evaluate the superfunction of zex.
The complex double implementation in C++ is loaded as
\url{https://mizugadro.mydns.jp/t/index.php/SuZex.cin}
\vskip 2mm

For real values of argument, the explicit plot of function SuZex is shown in figure
\ref{suzexplo} 
with thick line. For comparison, the thin line shows the zex function, id est, $y\!=\!z\exp(z)$. These curves cross at point $(1,\mathrm e)$ 
and in vicinity of point $(1.4,6.2)$.
\vskip 2mm

For real values of argument, SuZexp is positive monotonously growing function. At $-\infty$, it approaches zero, as the asymptotic representation
\rf{SuZexFM} prescribes. 

Then, the curve passes through point $(0,1)$, and then - through point $(1,\mathrm e)$.
At this point it grows a little bit slower, than function zex, but soon overdoes the zex, showing very fast growth; this growth is faster, than growth of any exponential.

\vskip 2mm

The same behaviour can be seen also at figure \ref{suzexmap}, that represents the complex map of function SuZex.
zex is entire function, id est, holomorphic in the whole complex plane. The inverse function, shown in 
figure \ref{suzexmap}, has cut. This inverse function
is considered in the next section.

\begin{figure}[t]
\sx{.34}{\begin{picture}(1252,1220)
\put(40,20){\ing{suzexmap}}
\put(18,1206){\sx{3.3}{$y$}}
\put(18,1113){\sx{3}{$5$}}
\put(18,1013){\sx{3}{$4$}}
\put(18, 913){\sx{3}{$3$}}
\put(18, 813){\sx{3}{$2$}}
\put(18, 713){\sx{3}{$1$}}
\put(18, 613){\sx{3}{$0$}}
\put(-6, 513){\sx{3}{$-1$}}
\put(-6, 413){\sx{3}{$-2$}}
\put(-6, 313){\sx{3}{$-3$}}
\put(-6, 213){\sx{3}{$-4$}}
\put(-6, 113){\sx{3}{$-5$}}
\put(-6, 013){\sx{3}{$-6$}}
\put(014, -5){\sx{3}{$-6$}}
\put(114, -5){\sx{3}{$-5$}}
\put(214, -5){\sx{3}{$-4$}}
\put(314, -5){\sx{3}{$-3$}}
\put(414, -5){\sx{3}{$-2$}}
\put(514, -5){\sx{3}{$-1$}}
\put(635, -5){\sx{3}{$0$}}
\put(735, -5){\sx{3}{$1$}}
\put(835, -5){\sx{3}{$2$}}
\put(935, -5){\sx{3}{$3$}}
\put(1035, -5){\sx{3}{$4$}}
\put(1135, -5){\sx{3}{$5$}}
\put(1227,-4){\sx{3}{$x$}}
\put(290,611){\sx{4}{$v\!=\!0$}}
\put(183,560){\sx{4}{\rot{90}$u\!=\!0.2$\ero}}
\put(468,560){\sx{4}{\rot{90}$u\!=\!0.4$\ero}}
\put(696,118){\sx{4}{\rot{83}$u\!=\!0$\ero}}
\put(980,236){\sx{4}{\rot{24}$u\!=\!-0.2$\ero}}
\put(790, 44){\sx{4}{\rot{38}$v\!=\!-0.2$\ero}}
\end{picture}}
\vskip -4pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:SuZexD1mapT.png}
\vskip -4pt
\caption{\large
$u\!+\!\mathrm i v =\mathrm{SuZex}(x\!+\!\mathrm i y)$
\label{suzexmap}}
\end{figure}

\begin{figure}[t]
\sx{.2}{\begin{picture}(2052,2020)
\put(100,100){\ing{auzexmap2}}
\put(40,2048){\sx{8}{$y$}}
\put(40,1880){\sx{7}{$8$}}
\put(40,1680){\sx{7}{$6$}}
\put(40,1480){\sx{7}{$4$}}
\put(40,1280){\sx{7}{$2$}}
\put(40,1080){\sx{7}{$0$}}
\put(-12,880){\sx{7}{$-2$}}
\put(-12,680){\sx{7}{$-4$}}
\put(-12,480){\sx{7}{$-6$}}
\put(-12,280){\sx{7}{$-8$}}
\put(002, 28){\sx{7}{$-\!10$}}
\put(220, 28){\sx{7}{$-8$}}
\put(420, 28){\sx{7}{$-6$}}
\put(620, 28){\sx{7}{$-4$}}
\put(820, 28){\sx{7}{$-2$}}
\put(1088, 28){\sx{7}{$0$}}
\put(1288, 28){\sx{7}{$2$}}
\put(1488, 28){\sx{7}{$4$}}
\put(1688, 28){\sx{7}{$6$}}
\put(1888, 28){\sx{7}{$8$}}
\put(2058, 28){\sx{7}{$x$}}
\put(258,1088){\sx{8}{\bf cut}}
\put(500,1840){\sx{8}{\rot{0}$v\!=\!0.6$\ero}}
\put(500,0310){\sx{8}{\rot{0}$v\!=\!-0.6$\ero}}
\put(444,0510){\sx{8}{\rot{62}$u\!=\!2$\ero}}
\put(1610,1312){\sx{8}{\rot{42}$v\!=\!0.2$\ero}}
\put(1770,1082){\sx{8}{$v\!=\!0$}}
\put(1650,0912){\sx{8}{\rot{80}$u\!=\!1.4$\ero}}
\put(1950,0802){\sx{8}{\rot{70}$u\!=\!1.6$\ero}}
\put(1650,0808){\sx{8}{\rot{-48}$v\!=\!-0.2$\ero}}
\end{picture}}
\vskip -4pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:AuZexMapT.jpg}
\vskip -8pt
\caption{\large
$u\!+\!\mathrm i v =\mathrm{AuZex}(x\!+\!\mathrm i y)$
\iL{auzexmap}}
\end{figure}

\section{AuZex}
\baselineskip 17.4pt
Complex map of function $\mathrm{AuZex}=\mathrm{SuZex}^{-1}$, id est, abelfunction for the transferfunction zex, is shown in the right hand side of figure \ref{auzexmap}. This section describes properties of function AuZex.
\vskip 2mm

The asymptotic expansion for the abelfunction for the transfer function AuZex can be obtained inverting the expansion of function SuZex, described in the previous section. However, one may consider as well the Abel equation for the abelfunction $G$ :

\be
G(\mathrm{zex}(z))=G(z)+1
\eL{AuZexGeq}

The asymptotics of solution $G=\mathrm{AuZex}$ can be expressed with
\be
G(z) \approx \frac{-1}{z}+\frac{1}{2}\ln(z)+\sum_{n=0}^N ~ b_n z^n +..
\ee
Coefficients $b_n$ for $n>1$ can be found substituting this expansion into the Abel equation \rf{AuZexGeq}.
This asymptotics provides the precise approximation for small values of $z$; at large values, the inverse of the Abel equation should be applied.
For some fixed $M$, let
\be
G_{M}(z) \approx \frac{-1}{z}+\frac{1}{2}\ln(z)+\sum_{m=0}^M ~ b_n z^n
\eL{zexGas}
and let 
\be
\mathrm{AuZex}(z) 
=\lim_{n\rightarrow \infty} F_M\Big(\mathrm{zex}^{-n}(z)\Big)+n
\nonumber \\
= \lim_{n\rightarrow \infty} F_M\Big(\mathrm{LambertW}^{\, n}(z)\Big)+n
\ee
Coefficient $C_0$ is chosen in such a way, that $\mathrm{AuZex}(1)=0$; then, the relation 
\be
\mathrm{SuZex}\Big(\mathrm{AuZex}(z)\Big)=z
\eL{SuAuZexz}
holds in wide range of values of $z$, except the negative part of the real axis.
With Mathematica software, the coefficients $b$ of the asymptotic expansion \rf{zexGas} can be calculated by the code below.
{\small \begin{verbatim}
zex[z_] = z Exp[z];
S[k_, L_] = Sum[a[k, m] L^m, {m, 0, k}]
F[K_, z_, L_] = Sum[S[k, L]/z^(k + 1), {k, 0, K}]
Series[zex[F[4,z,L]] - F[4, z+1,L+Log[1+1/z]], {z,Infinity,3}]
a[0,0] = -1; 
Series[zex[F[4, z, L]] - F[4, z+1, L+Log[1+1/z]], {z,Infinity,3}]
a[1, 1] = 1/2; a[1, 0] = 0; 
Simplify[Series[zex[F[5,z,L]] - F[5,z+1, L+Log[1+1/z]], {z,Infinity,4}]]

n = 2; 
s[n]=Series[zex[F[n+3,z,L]]-F[n+3,z+1,L+Log[1+1/z]],{z,Infinity,n+2}];
For[k = 0, k<=n,k++,m=n-k; 
a[n,m] = ReplaceAll[a[n, m],So1[Coefficient[s[n] L,L*L^m] == 0, a[n,m]]]; 
Print[n, Space, k, Space, m, Space, a[n, m] ] ]

n = 3;
s[n]=Series[zex[F[n+3,z,L]]-F[n+3,z+1,L+Log[1+1/z]],{z,Infinity,n+2}];
For[k = 0, k<=n,k++,m=n-k; 
a[n,m] = ReplaceAll[a[n, m],So1[Coefficient[s[n] L,L*L^m] == 0, a[n,m]]]; 
Print[n, Space, k, Space, m, Space, a[n, m] ] ]
\end{verbatim}} %Для нагладности, в этой програме нет цикла
and so on for higher 
$n$.
The first 9 coefficients are copypasted below:
\vskip 2mm

\be
\begin{array}{c|ccl}
n &  b_n & &{\rm ~approximation ~ of~ ~} b_n\\ \hline 
1 & -1/6 &\approx&-0.1666666666666666667\\
2 & 1/16 &\approx& ~ ~ \, 0.0625\\
3 & -19/540&\approx& -0.0351851851851851852 \\
4 & 1/48&\approx& ~ ~ \, 0.0208333333333333333 \\
5 & -41/4200&\approx& -0.0097619047619047619\\
6 & 37/103680&\approx& ~ ~ \, 0.00035686728395061728\\
7 & 18349/3175200&\approx&~ ~ \, 0.005778848576467624  \\
8 & -443/80640 &\approx&  -0.005493551587301587\\
9 &  55721/21555072&\approx&   -0.002585052835824441
\end{array}
\label{bArray}
\ee

The readers are invited to plot maps of ranges of validity of relations
\be
\mathrm{SuZex}(\mathrm{AuZex}(z))&=&z \\
\mathrm{AuZex}(\mathrm{SuZex}(z))&=&z 
\ee
These maps can be considered as confirmation, verification of the deduction above.
\vskip 2mm

\section{Iterates of zex}

With functions SuZex and AuZex, described in the previous sections, the iterates of function zex can be expressed as follows:
\be
\mathrm{zex}^n(z)=\mathrm{SuZex}\Big(n+\mathrm{AuZex}(z)\Big)
\eL{zexn}
As usually, the number $n$ of iterate has no need to be integer; it can be real or even complex.
\vskip 2mm

Iterates $y\!=\!\mathrm{zex}^n(x)$ versus $x$ for different $n$ are shown in figure \ref{zexiteplo}.
The integer iterates are shown with thick lines.
\vskip 2mm

The readers are invited to check the relation
\be
\mathrm{zex}^n\Big(\mathrm{zex}^m(z)\Big)=
\mathrm{zex}^{m+n}(z)
\eL{zexiteran}
and describe the range of validity in simple terms.
At least, this relation should hold in some vicinity of the positive part of the real axis for values of the parameters.
\vskip 2mm

Iterates of function zex are shown in figure \ref{zexiteplo}. They look similar to iterates of other fast-growing functions with real fixed point.
The iteration keeps the unity derivative in this point, so, all the curves in figure \ref{zexiteplo} approach the fixed point (id est, to the origin of coordinates) with unity derivative. Id est, with angle 45 degrees to the abscissa axis.
This property takes place also for other transfer functions with unity derivative at the fixed point.
One of such functions is considered in the next chapter.

\begin{figure}[t]
\vskip 8pt
\sx{.35}{\begin{picture}(1208,1208) 
\put(0,0){\ing{zexIteplo}} 
\put(11,1184){\sx{4.4}{$y$}} 
\put(04,1090){\sx{4}{$11$}} 
\put(04,990){\sx{4}{$10$}} 
\put(11,890){\sx{4}{$9$}} 
\put(11,790){\sx{4}{$8$}} 
\put(11,690){\sx{4}{$7$}} 
\put(11,590){\sx{4}{$6$}} 
\put(11,490){\sx{4}{$5$}} 
\put(11,390){\sx{4}{$4$}} 
\put(11,290){\sx{4}{$3$}} 
\put(11,190){\sx{4}{$2$}} 
\put(11,090){\sx{4}{$1$}} 
\put(91,6){\sx{4}{$1$}} 
\put(191,6){\sx{4}{$2$}} 
\put(291,6){\sx{4}{$3$}} 
\put(391,6){\sx{4}{$4$}} 
\put(492,6){\sx{4}{$5$}} 
\put(592,6){\sx{4}{$6$}} 
\put(693,6){\sx{4}{$7$}} 
\put(794,6){\sx{4}{$8$}} 
\put(894,6){\sx{4}{$9$}} 
\put(982,6){\sx{4}{$10$}} 
\put(1082,6){\sx{4}{$11$}} 
\put(1180,6){\sx{4.4}{$x$}} 
\put(96,1058){\sx{5}{\rot{88}$n\!=\!2$\ero}} 
\put(196,1058){\sx{5}{\rot{88}$n\!=\!1$\ero}} 
\put(168,706){\sx{4.5}{\rot{86}$y\!=\!\mathrm{zex}(x)$\ero}} 
\put(512,1030){\sx{5}{\rot{72}$n\!=\!0.3$\ero}} 
\put(629,1032){\sx{5}{\rot{64}$n\!=\!0.2$\ero}} 
\put(804,1037){\sx{5}{\rot{56}$n\!=\!0.1$\ero}} 
\put(1072,1052){\sx{5}{\rot{44}$n\!=\!0$\ero}} 
\put(772,752){\sx{5}{\rot{44}$y\!=\!x$\ero}} 
\put(1028,762){\sx{5}{\rot{32}$n\!=\!-0.1$\ero}} 
\put(1014,590){\sx{5}{\rot{23}$n\!=\!-0.2$\ero}} 
\put(1006,470){\sx{5}{\rot{17}$n\!=\!-0.3$\ero}} 
\put(510,128){\sx{4.3}{\rot{3}$y\!=\!\mathrm{LambertW}(x)$\ero}} 
\put(1010,164){\sx{5}{\rot{2}$n\!=\!-1$\ero}} 
\put(1010,65){\sx{5}{\rot{0}$n\!=\!-2$\ero}} 
\end{picture}}
\vskip 1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ZexIteT.jpg}
\vskip -2pt
\caption{\large $y\!=\!\mathrm{zex}^n(x)$ by  \rf{zexn} versus $x$ for various $n$ \iL{zexiteplo}}
\end{figure}
\vskip 2mm

I hope, the Readers can plot by themselves the complex maps of non-integer iterates of function zex.
The codes supplied to the figures above have the complex double implementation of 
SuZex and AuZex.

\chapter{~ Sin, super sin and iterates of sin}
\baselineskip 18pt

Not all exotic iterations can be constructed with formulas of the previous sections.
In the previous two sections, the transfer functions $T$ with fixed points $L$ are considered such that 
$T(L)\!=\!L$, $T'(L)\!=\!1$, $T''(L)\!>\!0$.
That deduction fails, if $T''(L)\!=\!0$. One of examples of such a transfer function is considered here.
\vskip 5pt

This chapter deals with transfer function 
$T\!=\!\sin$. 
This function is often used in various applications, so, I think, it deserves a special chapter.
\vskip 5pt

Iterates of sin had been considered since century 19, 
but the rough approximations had been suggested only in 2011 \cite{kur,car}. Then in 2014, the efficient approximation had been reported in the 
Far East Journal of Mathematical Science \cite{susin}. Below, I retell the key ideas of that publication.
\vskip 5pt


I hope, the Reader can plot the complex maps of sin and arcsin. I recommend that Reader does this as an exercise. And also, the explicit plots of these functions. After to watch the pictures mentioned, one can understand the sense of superfunction of sin, I call it SuSin. 
SuSun is solution of the transfer equation, I repeat is once again,
\vskip .1pt
\hskip 44pt
$\sin\!\big(\mathrm{SuSin}(z)\big)\!=\!\mathrm{SuSin}(z\!+\!1)$
\vskip .1pt
Explicit plot of function SuSin of real argument is shown in figure  \ref{susinplot}. In the next section, I describe the construction of this function.

\begin{figure}[b]
\sx{.412}{\begin{picture}(1016,208)
\put(20,1){\includegraphics{susinplot}}
\put(2,197){\sx{2.4}{$y$}}
\put(-1,151){\sx{2.8}{$\frac{\pi}{2}$}}
\put(2,93){\sx{2.4}{$1$}}
\put(2,-5){\sx{2.4}{$0$}}
\put(15,-22){\sx{2.4}{$0$}}
\put(115,-22){\sx{2.4}{$1$}}
\put(215,-22){\sx{2.4}{$2$}}
\put(315,-22){\sx{2.4}{$3$}}
\put(415,-22){\sx{2.4}{$4$}}
\put(516,-22){\sx{2.4}{$5$}}
\put(616,-22){\sx{2.4}{$6$}}
\put(717,-22){\sx{2.4}{$7$}}
\put(817,-22){\sx{2.4}{$8$}}
\put(917,-22){\sx{2.4}{$9$}}
\put(1010,-22){\sx{2.5}{$x$}}
\put(140,166){\sx{2.8}{$y\!=\! \sqrt{3/x}$}}
\put(135,56){\sx{2.8}{$y\!=\!\mathrm{SuSin}(x)$}}
\end{picture}}
\vskip 6pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Susinplot.jpg}
\vskip -2pt
\caption{\large $y\!=\!\mathrm{SuSin}(x)$ 
by \rf{susindef} and its asymptotics
\rf{susin1o}  \iL{susinplot} }
\end{figure}



\section{Super sin}
\baselineskip 21pt

Superfunction of sin, called SuSin, is shown in figure 
\ref{susinplot}. It is solution of the transfer equation with sin as transfer function; I repeat it once again:
\be
F(z\!+\!1)=\sin\Big(F(z)\Big)
\eL{Fz1sinFz}
In this section, the special solution $F=\mathrm {SuSin}$ is constructed with the following properties:
\be
\mathrm{SuSun}(0)&=&\pi/2   ~ ~ ~ ~ ~ \iL{susinp2}\\
\mathrm{SuSun}(z)&=&\sqrt{\frac{3}{z}}\left(1+O\!\left(\frac{\ln(z)}{z}\right)\right)
\eL{susin1o}

The leasing term of the right hand side of \rf{susin1o} can be guessed replacing $F(z\!+\!1)$ to $F(z)+F^\prime(z)$ in the left hand side of equation \rf{Fz1sinFz} and solving the resulting differential equation. However, there is certain residual at the substitution of such an approximation into \rf{Fz1sinFz}; and this residual indicates the form of the next term in expansion \rf{susin1o}. Expression $\ds \frac{\ln(z)}{z}$ appears as the effective small parameter of the expansion. The residual at the substitution of representation \rf{susin1o} into the transfer equation
\rf{Fz1sinFz} helps to guess a form of the higher term of the expansion, and so on. After few such steps, I guess and verify the following form of the  
approximation for the superfunction $F$ of sin:
\be
F(z)=F_M(z)+O\left( \frac{\ln(z)^{M+1}}{z^{M+3/2}} \right)
~ ~ ~ \eL{susinasymf}
where
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F_M(z)=\sqrt{\frac{3}{z}}\left(1-\frac{3}{10} \frac{\ln(z)}{z}+\sum_{m=2}^M P_m(\ln(z)) \, z^{-m} 
\right)
\eL{susinFM}
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
P_M(z)=\sum_{n=0}^m a_{m,n} (-z)^m ~~~~~~~~~
\eL{susina}
and coefficients $a$ are constants. These constants can be calculated with the Mathematica code below:
\baselineskip  19pt
{\small \baselineskip 15pt
\begin{verbatim}
P[m_, L_] := Sum[a[m, n] L^n, {n, 0, m}]
F[z_] = Sqrt[3/z] ( 1 + Sum[P[m, Log[z]]/z^m, {m, 1, M}])
M = 9; a[1, 0] = 0;
F1x = F[1 + 1/x];
Ftx = Sin[F[1/x]];
s[1] = Series[(F1x - Ftx)/Sqrt[x], {x, 0, 2}]
t[1] = Extract[Solve [Coefficient[s[1], x^2] == 0, {a[1, 1]}], 1]
A[1, 1] = ReplaceAll[a[1, 1], t[1]]
su[1] = t[1]

m=2;
s[m]=Simplify[ReplaceAll[Series[(F1x-Ftx)/Sqrt[3 x],{x,0,m+1}], su[m-1]]]
t[m] = Simplify[Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m+1)]]
u[m] = Simplify[Collect[t[m], L]]
v[m] = Table[Coefficient[u[m] L, L^(n+1)] == 0, {n, 0, m}]
w[m] = Table[a[m, n], {n, 0, m}]
ad[m] = Extract[Solve[v[m], w[m]], 1]
su[m] = Join[su[m - 1], ad[m]]

m=3;
s[m]=Simplify[ReplaceAll[Series[(F1x-Ftx)/Sqrt[3 x],{x,0,m+1}],su[m-1]]]
t[m] = Simplify[Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m+1)]]
u[m] = Simplify[Collect[t[m], L]]
v[m] = Table[Coefficient[u[m] L, L^(n+1)] == 0, {n, 0, m}]
w[m] = Table[a[m, n], {n, 0, m}]
ad[m] = Extract[Solve[v[m], w[m]], 1]
su[m] = Join[su[m - 1], ad[m]]
. . .
\end{verbatim}
}
and so on. The resulting coefficients are shown in table \ref{tablesusin}.

\begin{table}[b]
\caption{Table of coefficients $a$ in equation \rf{susina}   \label{tablesusin}}
\hskip -4pt
$\begin{array}{ccccccccc}
a_{1,0} & \frac{3}{10} & a_{1,2} & a_{1,3} & a_{1,4} &  a_{1,5} & a_{1,6} & a_{1,7} & %a_{1,8} 
\rule{0pt}{13pt}\\
\frac{79}{700} & \frac{9}{50} & \frac{27}{200} & a_{2,3} & a_{2,4} & a_{2,5} & a_{2,6} & a_{2,7}
\rule{0pt}{14pt}\\
\frac{411}{3500} & \frac{1941}{7000} & \frac{27}{125} & \frac{27}{400} & a_{3,4} & a_{3,5} & a_{3,6} & a_{3,7}
\rule{0pt}{14pt}\\
\frac{1606257}{10780000} & \frac{7227}{17500} & \frac{1683}{4000} & \frac{1917}{10000} & \frac{567}{16000} & a_{4,5} & a_{4,6} & a_{4,7}
\rule{0pt}{14pt}\\
\frac{140345627}{700700000} & \frac{70079931}{107800000}  & \frac{566973}{700000} & \frac{98739}{200000} & \frac{7533}{50000} & \frac{15309}{800000} & a_{5,6} &
a_{5,7}%& a_{5,8}
 \rule{0pt}{14pt}\\
\frac{137678711441}{490490000000} & \frac{7364523}{7007000} & \frac{305491257}{196000000} & \frac{4155111}{3500000} & \frac{796311}{1600000} & \frac{2218347}{20000000} & \frac{168399}{16000000} &  a_{6,7}
\rule{0pt}{14pt}\\
\frac{25317035192599}{62537475000000} \!\!&
\frac{8462569406199}{4904900000000} \!\!&
\frac{32174780481}{10780000000} \!\!&
\frac{5367503637}{1960000000} \!\!&
\frac{407711313}{280000000} \!\!&
\frac{181900809}{400000000} \!\!&
\frac{1960281}{25000000} \!\!&
\frac{938223}{160000000}
\rule{0pt}{14pt}
\JP{
\\
\frac{20390161579066727223}{34157135012000000000} &
\frac{1191102095494119}{416916500000000} &
\frac{111297984941043}{19619600000000} &
\frac{164937331809}{26950000000} &
\frac{310033164249}{78400000000} &
\frac{11083463037}{7000000000} &
\frac{3104886087}{8000000000} &
\frac{301983147}{5600000000} &
\frac{8444007}{2560000000}  \rule{0pt}{14pt}
}
\end{array}$
\end{table}
\vskip 2mm

For some positive integer $M$, define function $F$ with
\be
F=\lim_{k \rightarrow \infty} \arcsin^k(F_M(z+k))
\eL{SuSinF}
The resulting $F$ does not depend on the number $M$ of terms taken into account in the expansion \rf{susinFM}. However, the larger, $M$, the faster is convergence of the limit in \rf{SuSinF}. This gives efficient algorithm for evaluation of superfunction $F$ for the transfer function $\sin$.
\vskip 2mm

\begin{figure}[t]
\sx{.2}{ \begin{picture}(2016,2044)
\put(48,40){\includegraphics{susinmap}}
\put(4,2018){\sx{5.4}{$y$}}
\put(4,1824){\sx{5}{$8$}}
\put(4,1624){\sx{5}{$6$}}
\put(4,1424){\sx{5}{$4$}}
\put(4,1224){\sx{5}{$2$}}
\put(4,1024){\sx{5}{$0$}}
\put(-28,824){\sx{5}{$-2$}}
\put(-28,624){\sx{5}{$-4$}}
\put(-28,424){\sx{5}{$-6$}}
\put(-28,224){\sx{5}{$-8$}}
\put(-8,-9){\sx{5}{$-10$}}
\put(200,-8){\sx{5}{$-8$}}
\put(400,-8){\sx{5}{$-6$}}
\put(600,-8){\sx{5}{$-4$}}
\put(800,-8){\sx{5}{$-2$}}
\put(1040,-8){\sx{5}{$0$}}
\put(1241,-8){\sx{5}{$2$}}
\put(1442,-8){\sx{5}{$4$}}
\put(1643,-8){\sx{5}{$6$}}
\put(1843,-8){\sx{5}{$8$}}
\put(2028,-8){\sx{5.2}{$x$}}
\put(1436,930){\sx{7}{\rot{89}$u\!=\!0.7$\ero}}
\put(1640,930){\sx{7}{\rot{89}$u\!=\!0.6$\ero}}
\put(1980,930){\sx{7}{\rot{89}$u\!=\!0.5$\ero}}
\put(1520,1660){\sx{7}{\rot{68}$v\!=\!-0.2$\ero}}
\put(1790,1406){\sx{7}{\rot{36}$v\!=\!-0.1$\ero}}
\put(250,1030){\sx{5.2}{\bf cut}}
\put(928,1148){\sx{7}{\rot{0.}$u\!=\!1$\ero}}
\put(668,1080){\sx{7}{\rot{0.}$v\!=\!-1$\ero}}
\put(688,960){\sx{7}{\rot{0.}$v\!=\!1$\ero}}
\put(1680,1024){\sx{7}{$v\!=\!0$}}
\put(1740,664){\sx{7}{\rot{-35}$v\!=\!0.1$\ero}}
\put(1478,418){\sx{7}{\rot{-68}$v\!=\!0.2$\ero}}
\end{picture}}
\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Susinmap.jpg}
\vskip -6pt
\caption{\large
$u\!+\!\mathrm i v=\mathrm{SuSin}(x\!+\!\mathrm i y)$ by \rf{susindef} \label{susinmap} }
\end{figure}
For superfunction, declared in the beginning of the section, id est, satisfying equation \rf{susinp2}, define
\be\!\!\!
\mathrm{SuSin}(z)=F(z\!+\!x_1)
\label{susindef} \ee
where $x_1\!\approx\! 1.4304553465288~$ is solution of equation 
\be \!\!\!\!
F(1\!+\!x_1)=1
\label{susinx1}\ee
Note that the required value at zero is achieved, because
$\mathrm{SuSin}(0)\!=\!\arcsin(\mathrm{SuSin}(1))\!=\!\arcsin(1)\!=\!\pi/2$.

\vskip 2mm
Function SuSin is shown as explicit plot and as complex map in figures \ref{susinplot} and \ref{susinmap}.
Function SuSin has sqrt-type singularity as zero, and it has cut line along the negative part of the real axis. In the rest of the complex plane, SuSin is holomorphic.
At infinity,  SuSin slowly decays to zero, according to its asymptotic \rf{susin1o}. This asymptotic is shown in figure \ref{susinplot} with thin line. 
\vskip 2mm


In this Book, SuSin is first example of superfunction, for which the transfer function shows growth slower than linear. In this case, for positive values of the argument (id est, larger than the fixed point), the superfunction is monotonically decreasing, as it is seen in Figure \ref{susinplot}. One could expect the inverse function (Abel sin) to decrease, at least for some moderate positive values of the argument. This inverse function is considered in the next section.

\section{AuSin}
\baselineskip 18.4pt

\begin{wrapfigure}{r}{210px}
\normalsize
\vskip -46pt
\sx{.6}{\begin{picture}(320,431)
\put(14,16){\includegraphics{ausinplo}}
\put(1,410){\sx{2.2}{$y$}}
\put(0,308){\sx{2.2}{$3$}}
\put(0,208){\sx{2.2}{$2$}}
\put(0,108){\sx{2.2}{$1$}}
\put(0,8){\sx{2.2}{$0$}}
\put(10,-4){\sx{2.2}{$0$}}
\put(110,-4){\sx{2.2}{$1$}}
\put(151,-4){\sx{2.1}{$\pi/2$}}
\put(210,-4){\sx{2.2}{$2$}}
\put(320,-3){\sx{2.2}{$x$}}
\put(348,19){\rot{90}\sx{1.08}{\url{https://mizugadro.mydns.jp/t/index.php/File:Ausinplot1.jpg}}\ero}
\end{picture}}
\vskip 1pt
\caption{\large
$y\!=\!\mathrm{AuSin}(x)$ \label{ausinplot}}
\vskip 12pt
\end{wrapfigure}

Here, the inverse function of super sin is called AuSin; its explicit plot is shown in figure \rf{ausinplot}. In this chapter, I describe evaluation of this function.
\vskip 2mm

For the sinusoidal transfer function, the Abel function $G\!=\!F^{-1}$ satisfies the Abel equation
 \be
\!\!\!\!\!G(\sin(x))=G(z)\!+\!1
\label{abelsineq}
\ee
From the properties of SuSin, the properties of the inverse function can be guessed. In particular, for large values of the argument, the asymptotic  below should hold:
\be
G(z)=\frac{2}{z^2} +O(\ln(z))
\eL{AbelsinGas}

\vskip 2mm
Abel sin can be constructed in analogy with super sin. The solution $G$ of the Abel equation \rf{abelsineq} with the asymptotic representation
\rf{AbelsinGas} can be constructed, inverting function $F$ by \rf{SuSinF}, id est,  $G\!=\!F^{-1}$. 
Then, the constant $x_1$ should be added, 
\be
\mathrm{AuSin}(z)=G(z)+x_1
\label{ausinG}
\ee
in order to satisfy relation
\be
\mathrm{AuSin}(\pi/2)=0
\eL{ausinp2}
\vskip 2mm

First, construct the approximation of function $G$; let
\be\!\!\!\!\!\!
G_M(z)\!=\!\frac{3}{z^2}\!+\!\frac{5}{6}\ln(z)\!+\!\!\sum_{m=1}^M c_m z^{2m} ~ ~
\label{sinGM}\ee
Subsitution of $g(z) = G_M+O(z^{2M+2})$ into the Abel equation \rf{abelsineq} gives the coefficients $c$. In particular, 
\be
c_1&\!=\!& ~ ~ \frac{79}{1050} \\
c_2&\!=\!& ~ ~ \frac{29}{2625} \rule{0pt}{19pt}\\
c_3&\!=\!&\frac{91543}{36382500}\rule{0pt}{19pt}\\
c_4&\!=\!&\!\!\frac{18222899}{28378350000} \rule{0pt}{19pt}
\label{abelisinc}
\ee
Then, for some fixed $M$,  function $G$ can be expressed as limit
\be\!\!\!\!\!
G(z)\!=\!\lim_{k\rightarrow \infty} G_M(\sin^k(z))-k
\eL{singlim}
and $\mathrm{AuSin}\!=\!\mathrm{SuSin}^{-1}$ appears as %can be expressed as follows:
\be
\mathrm{AuSin}(z)=G(z)-G(\pi/2)
\eL{AuSinDe}
Term
$G(\pi/2)\!\approx\! 2.089622719729524$ in equation \rf{AuSinDe} 
provides condition \rf{ausinp2}.
\vskip 2mm

 %\be
Complex map of abelsinus AuSin by \rf{AuSinDe} is shown in fgure \ref{ausinmap}. 
As for other real-holomorphic functions, this map is symmetric with respect to reflection from the real axis, id est, with respect to flipping upside-down.
In addition, the map is symmetric with repeat to reflections from the axis 
$x\!=\!\pi/2$; 
the first evaluation of function sin in formula
\rf{singlim} 
provides this symmetry.
\vskip 2mm

For the central part of figure \ref{ausinmap}, the limit \rf{singlim} 
converges and defines the holomorphic function in a pretty regular way.
At lateral parts of figure \ref{ausinmap}, the lines of level of the real part and those of the imaginary part form the fractal-like structures.
There, the AuSin cannot be considered as inverse function of SuSin. Actually, the range of validity of the inverse function is even narrower.
\newpage

\begin{figure}[h]
\hskip 8pt
\sx{1.28}{\begin{picture}(308,420)
\put(10,9){\includegraphics{ausinmappng}} %extended
\put(2,406){\sx{1.2}{$y$}}
\put(2,306){\sx{1.2}{$1$}}
\put(2,206){\sx{1.2}{$0$}}
\put(-7,106){\sx{1.2}{$-1$}}
\put(9,-2){\sx{1.2}{$0$}}
\put(109,-2){\sx{1.2}{$1$}}
\put(157,-2){\sx{1.2}{$\pi/2$}}
\put(209,-2){\sx{1.2}{$2$}}
\put(309,-2){\sx{1.2}{$3$}}
\put(319,-2){\sx{1.2}{$x$}}
\put(184,208){$v\!=\!0$}
\put(98,200){\rot{90}$u\!=\!2$\ero}
\put(114,200){\rot{90}$u\!=\!1$\ero}
\put(147,186){\rot{41}$u\!=\!0$\ero}
\put(136,131){\rot{-40}$u\!=\!-1$\ero}
\put(125,93){\rot{23}$v\!=\!0$\ero}
\put(92,288){\rot{66}$u\!=\!-2$\ero}
\put(131,282){\rot{31}$u\!=\!-1$\ero}
\put(148,260){$u\!=\!-0.5$}
\put(86,158){\rot{90}$v\!=\!3$\ero}
\put(100,155){\rot{90}$v\!=\!2$\ero}
\put(122,148){\rot{90}$v\!=\!1$\ero}
\put(159,144){\rot{90}$v\!=\!0.2$\ero}
\put(172,144){\rot{90}$v\!=\!0$\ero}
\put(186,138){\rot{90}$v\!=\!-0.2$\ero}
\end{picture}}
\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Ausinmap52r5.jpg}
\vskip -4pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{AuSin}(x\!+\mathrm i y)$  by formula \rf{AuSinDe}~ \iL{ausinmap}}
\vskip -80pt
\end{figure}

\rule{0pt}{80pt}

Range of validity of relation
\be
\mathrm{SuSin}\Big(\mathrm{AuSin}(z)\Big)=z
\eL{SuSinA}
is 
shown in figure \ref{susinrange}, this range is shaded with rectangular grid.

\begin{figure}[h]
\hskip 8pt
\sx{1.28}{\begin{picture}(308,420)
\put(10,9){\includegraphics{susinau}}
\put(10,9){\includegraphics{susinau2}}
\put(2,406){\sx{1.2}{$y$}}
\put(2,306){\sx{1.2}{$1$}}
\put(2,206){\sx{1.2}{$0$}}
\put(-7,106){\sx{1.2}{$-1$}}
\put(9,-1){\sx{1.2}{$0$}}
\put(109,-1){\sx{1.2}{$1$}}
\put(157,-1){\sx{1.2}{$\pi/2$}}
\put(209,-1){\sx{1.2}{$2$}}
\put(309,-1){\sx{1.2}{$3$}}
\put(319,-1){\sx{1.2}{$x$}}
\end{picture}}
\vskip 2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Ausinsusinmapt50.jpg}
\vskip -8pt
\caption{$u\!+\!\mathrm i v=h(x\!+\!\mathrm i y)$  and lines $\Im(\mathrm{AuSin}(x\!+\!\mathrm i y)\!=\!0$  ~ \label{susinrange}}
\vskip -200pt
\end{figure}
\rule{0pt}{190pt}

\baselineskip 18pt
Technically, the shading in figure \ref{susinrange}
is built up as the complex map of function
\be 
h(z)=\mathrm{SuSin}\Big(\mathrm{AuSin}(z)\Big)
\label{susinausin}
\ee
In addition, in figure \ref{susinrange}, lines of level $\Im(\mathrm{AuSin}(x+\mathrm i y))$ are shown; they are borrowed from figure 
\ref{ausinmap}. These lines bound the range of validity of relation $h(z)=z$.

The readers are cordially invited to download the generator of the figure and modify it, to plot the map of the agreement function
\be
A(z)=-\lg\left( \frac{|h(z)-z|}{|h(z)|+|z|}\right)
\iL{aausinsu}\ee
and to verify, that in the shaded part, relation $\mathrm{AuSin}=\mathrm{SuSin}^{-1}$ holds with at least 14 decimal digits.
At the right hand side of the figure \ref{susinrange}, relation  $\mathrm{AuSin}=\mathrm{SuSin}^{-1}$ is not valid; $\mathrm{AuSin}(x\!+\!\mathrm i y)$ is symmetric with respect to line $x\!=\!\pi/2$; so, it cannot distinguish values for the right hand side in figure \ref{ausinplot} from those at the left hand side.
\vskip 2mm

\begin{wraptable}{r}{134pt}
\vskip -14pt
\caption{%Coefficients $b$ in \rf{ausintaylor} 
\label{ausintaytab}}
\vskip -9pt
$ ~ 
\begin{array}{cc}
n & b_n\\
1 & 2.29163807440958 \\
2 & 1.96043852439688 \\
3 & 1.07862851256147 \\
4 & 0.59622997993395 \\
5 & 0.28333997139829 \\
6 & 0.14193261194548 \\
7 & 0.06423734271234 \\
8 & 0.03026687705508 \\
9 & 0.01351721250427
\end{array}
$
\vskip -12pt
\end{wraptable}

For the efficient (fast and precise) numerical implementation of AuSin, various expansions can be used.
The Taylor expansion at $\pi/2$ can be written as follows:
\be \!\!\!\!\!\!\!\!\!
\mathrm{AuSin}\left( \frac{\pi}{2}+t \right) =
\sum_{n=1}^{\infty} b_n t^{2n}
\label{ausintaylor}
\ee
Coefficients $b$ in the expansion \rf{ausintaylor} are evaluated with 
representation of AuSin through function $G$ by \rf{sinGM}, \rf{singlim}, \rf{AuSinDe}.
Approximations of first nine of these coefficients are shown in Table \ref{ausintaytab}.
\vskip 2mm

Series \rf{ausintaylor} converges at $|t|\!<\!\pi/2$. 
It is sufficient to take few terms of this expansion in order to reproduce the most of map in figure \ref{ausinmap}. 
Readers are invited to plot this map (or, at least, to look for it in the Appendix, figure 
\ref{ausinte} at page \pageref{ausinte}).%\pageref{ausintee}
\vskip 2mm


Taking into account some tens of coefficients in series \rf{ausintaylor}, 
the numerical approximations of AuSin provides the precision at least not worse, than the precision of the original representation through the asymptotic formulas \rf{sinGM}, \rf{singlim}, \rf{AuSinDe}.
However the original representation is still necessary to calculate the coefficients of the secondary expansions, and, in particular, those of the Taylor expansion 
\rf{ausintaylor}.
Optimisation of such representations may have sense before to include them to some software as built-in functions, while the each microsecond at the evaluation is important for the resulting efficiency at the multiple evaluations. However, even with the primitive approximations described above, 
the functions are evaluated with approximately 14 decimal digits and allow to plot complex maps in real time.
This indicates the good efficiency of the representations suggested.

\section{Iterated sin}
\baselineskip 17.2pt
With functions $\mathrm{SuSin}$ and $\mathrm{AuSin}=\mathrm{SuSin}^{-1}$, defined above, the iterates of sin can be expressed as follows:
\be
\sin^n(z)=\mathrm{SuSin}\Big(n+\mathrm{AuSin}(z)\Big)
\eL{sinn}
This formula looks pretty similar to representations of iterates of any other function with determined superfunction and the corresponding Abel function.
In this representation, number $n$ of iterates has no need to be integer; it can be real and even complex.
For real values of number $n$ of iterates, $\sin^n(x)$ is plotted versus $x$  in figure \ref{sinite}.

\begin{figure}[t]
\hskip 6pt
\sx{1.3}{\begin{picture}(328,168)
\put(4,9){\includegraphics{sinite}}
\put(-1.6,162){\sx{1.001}{$y$}}
\put(-1.6,106){\sx{1.001}{$1$}}
\put(-1.6, 06){\sx{1.001}{$0$}}
\put(3,0){\sx{1.001}{$0$}}
\put(103,0){\sx{1.001}{$1$}}
\put(153,0){\sx{1.001}{$\pi/2$}}
\put(203,0){\sx{1.001}{$2$}}
\put(303,0){\sx{1.001}{$3$}}
\put(315,0){\sx{1.001}{$x$}}
\put(24.6,115){\sx{.8}{\rot{89}$n\!=\!-100$\ero}}
\put(42.6,115){\sx{.8}{\rot{89}$n\!=\!-20$\ero}}
\put(53,115){\sx{.8}{\rot{87}$n\!=\!-10$\ero}}
\put(67,115){\sx{.8}{\rot{85}$n\!= -5$\ero}}
\put(77,115){\sx{.8}{\rot{82}$n\!=\!-3$\ero}}
\put(84.3,115){\sx{.8}{\rot{79}$n\!=\!-2$\ero}}
\put(94,112){\sx{.8}{\rot{68}$n\!=\!-1$\ero}}
\put(123.4,134){\sx{.8}{\rot{63}$n\!=\! -0.2$\ero}}
\put(146,147){\sx{.9}{\rot{45}$n\!=\!0$\ero}}
\put(142,140){\sx{.8}{\rot{10}$n\!= 0.1$\ero}}
\put(140,131.6){\sx{.8}{\rot{10}$n\!= 0.2$\ero}}
\put(139,122){\sx{.8}{\rot{9}$n\!= 0.4$\ero}}
\put(110,95){\sx{.8}{\rot{21}$n\!= 1$\ero}}
\put(110,84.3){\sx{.8}{\rot{13}$n\!= 2$\ero}}
\put(110,77.3){\sx{.8}{\rot{8}$n\!= 3$\ero}}
\put(110,67){\sx{.8}{\rot{6}$n\!= 5$\ero}}
\put(110,53){\sx{.8}{$n\!=\! 10$}}
\put(110,42.3){\sx{.8}{$n\!= 20$}}
\put(110,24.3){\sx{.8}{$n\!=\!100$}}
\end{picture}}
\vskip -4pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sinplo2t100.jpg}
\vskip -8pt
\caption{\large $y\!=\! \sin^n(x)$ by \rf{sinn}  versus $x$ for various $n$ \iL{sinite}}
\end{figure}
\vskip 2mm

For positive number $n$ of iterate, graphics of $y=\sin^n(x)$ are symmetric with respect to line $x\!=\!\pi/2$. The larger is $n$, the closer the curve approaches the abscissa axis. 
\vskip 6pt

For negative $n$, the graphic reaches the branch point at $y=\pi/2$ and cannot be continued be continued above, as the iterates get complex values. A usually, the 0th iterate corresponds to the identity function, $\sin^0(x)=x$, and this relation holds while $x<\pi/2$.
\vskip 6pt

Through the iterates \rf{sinn} of sin, the SuSin can be expressed as follows:
\be
\mathrm{SuSin}(z)= \sin^z(\pi/2)
\eL{susinzsin} 

From the point of view of computation, representation  \rf{susinzsin} does not have much sense.
Anyway, for the evaluation of the right hand side of  \rf{susinzsin}, the approximations of SuSin should be used.
However, such a representation may have sense at the building-in of superfunctions (and non-integer iterates) into the programming languages.
\section{Application}
\baselineskip 19pt
In this section, I discuss applications of multiple iterates of sin. However, I assume, that 
the Readers can freely use SuSin and AuSin, described above, to plot new graphics.
I load the complex double implementations of these functions as \\
\url{https://mizugadro.mydns.jp/t/index.php/Susin.cin} and \\
\url{https://mizugadro.mydns.jp/t/index.php/Ausin.cin} \\
Beautiful figures can be plotted playing with SuSin and AuSin.
\vskip 2mm

For the physical applications, the real number of iterate is simpler to interpret, than the complex iterate. Keeping in mind the application, I suggest the example of parameterisation of the shape of the sled runner with high iterate of sin. Figure \ref{boy} shows 
\be
y =\sin^n(\pi/2)\!-\!\sin^n(x)
\ee
plotted over the photo of the sled (with a boy in it), taken from Wikimedia Common
\cite{boy}. 

The curve, that overlaps wight he sled runner, is the 100th iterate of sin; the number of iterate $n\!=\!100$ is the only adjusted parameter used for the fitting.
However, the photo is shifted, scaled and rotated, in order to have the tip at $x=0$ and the last support of the sled runner at $x\!=\!\pi/2$, $y\!=\!0$;
in this point the sled runner is horizontal, this is provided by the slight rotation of the photo  (for one degree).

~

\begin{figure}[h]
\hskip 30pt
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\put(14,9){\includegraphics{boy}}
\put(3,28){\sx{1.001}{$y$}}
\put(0,16){\sx{1.001}{$0.1$}}
\put(0, 06){\sx{1.001}{$0$}}
\put(13,0){\sx{1.001}{$0$}}
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\put(213,0){\sx{1.001}{$2$}}
\put(224,0){\sx{1.001}{$x$}}
\end{picture}}
\vskip -4pt
\hskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Boyt100.jpg}
\vskip -4pt
\caption{Boy at the sledge and $y\!=\!\sin^n(\pi/2)\!-\!\sin^n(x)~$ for $~n\!=\!100$ \label{boy}}
\vskip -4pt
\end{figure}
\vskip 2mm


\chapter{~ Nemtsov function}
\baselineskip 18pt
\begin{wrapfigure}{r}{133px}
\sx{.3}{
\begin{picture}(404,388)
\put(2,2){\ing{nemplo}}
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\end{picture}}
\vskip -9pt
\caption{$y\!=\!\mathrm{Nem}_q(x)$ %for various $q$ 
\label{fignem} }
\end{wrapfigure}

This chapter represents the last (for this book) example of exotic iteration, and deal with the specific polynomial
\be\!\!\!\!\!\!\!\!
T(z)=\mathrm{Nem}_q(z)=z+z^3+qz^4
\label{nem}
\ee
I assume, that $q$ is positive real parameter. For various values of $q$, the explicit plot of this function is shown in figure 
\ref{fignem}. This is simple, but still non-trivial case of the transfer function $T$, for which 
\be\!\!\!\!\!
T(L)\!=\!L ~,~~
T'(L)\!=\!1 ~,~~
T''(L)\!=\!0 ~,~~
T'''(L)\!>\!0 ~,~~
T''''(L)\!>\!0 ~
\ee
Note, that for $T\!=\!\sin$, considered in the previous section, $L\!=\!0$ and $T''''(L)\!=\!0$; function sin is antisymmetric, $T(-z)\!=\!-T(z)$, and, in this sense, is simpler, than 
$T\!=\!\mathrm{Nem}_q$. 
\vskip 4pt

The analysis of transfer function {\ref{nem}) is my last attempt (up to year 2020) to write an elementary function, for which I  would not be able construct superfunction, abelfunction and the non-integer iterates. 
I found, that I need a special name for function by \rf{nem}. That happened 2015.01.28; the same day, murder of Boris Nemtsov had been reported
\footnote{\url{https://mizugadro.mydns.jp/t/index.php/Putin_killed_Nemtsov}};
so I pick up the first three letters of his last name to denote the function.
\vskip 5pt

In this Book, I do not follow the historical timeline of events, so, I put chapter about the Nemtsov function here, in order to have the exotic iterations in one bunch.
For real argument, graphics of the Nemtsov function for various values of parameter $q$ are shown in figure 
\ref{fignem}. Complex maps of function $\mathrm{Nem}_q$ and its inverse function $\mathrm{ArqNem}_q\!=\!\mathrm{Nem}_q^{-1}$ 
are shown in figure \ref{nema}
for 
$q\!=\!0$,
$q\!=\!1$ and
$q\!=\!2$. Function ArqNem is described in the following section.

\begin{figure}[tp]
\normalsize
\parindent 0pt

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\vskip -40pt
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\vskip 12pt

\hskip -4pt
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\vskip 12pt

\hskip -4pt
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\vskip -4pt
\caption{
Maps
$u\!+\!\mathrm i v=\mathrm{Nem}_q(x\!+\!\mathrm i y)$ for $q\!=\!0,1,2$, left column,  and 
maps
$u\!+\!\mathrm i v=\mathrm{ArqNem}_q(x\!+\!\mathrm i y)$, right column, for the same $q$
\label{nema}
}
\end{figure}

\section{ArqNem}
\baselineskip 17.8pt
For the efficient evaluation of iterates of a function, we need both, its superfunction and the abelfunction.
For implementations of these functions, we need both the transfer function $T\!=\!\mathrm{Nem}{_q}$ and its inverse $\mathrm{ArqNem}_q\!=\!T^{-1}$.
For real values of the argument, plot $y=\mathrm{ArqNem}_q(x)$
for 
$q\!=\!0$,
$q\!=\!1$,
$q\!=\!2$,
$q\!=\!4$,
$q\!=\!10$
is shown in figure \ref{arqnemplo}.
Inversion of the Nemtsov function happens to be not trivial, and its description deserves the special section. 
\vskip 6pt

\begin{figure}[t]
\vskip -22pt
\sx{.8}{\url{https://mizugadro.mydns.jp/t/index.php/File:Arqnemplot.jpg}}

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\end{picture}}
\vskip -12pt
\caption{$y\!=\!\mathrm{ArqNem}_q(x)$ for $q\!=\!0,1,2$
\iL{arqnemplo}}
\vskip 18pt
\end{figure}

The Nemtsov function $\mathrm{Nem}_q$ by \rf{nem} is the 4th order polynomial; for $q\!>\!0$, equation
\be
\mathrm{Nem}_q(x)=z
\eL{Nemqxz}
at given $z$ has four solutions $x$. Any of original solutions (suggested, for example, by Mathematica routine {\bf Solve})
happen to be ugly at the complex map, and even worse at evaluation of abelfunction for the Nemtsov function.
The reader can plot the complex maps for the four roots of equation \rf{Nemqxz} and 
see the root of my discontent with them. But these solutions can be used to construct the inverse function shown the right hand column of figure \ref{nema}.
I have implemented several combinations of the ``primary'' solutions,  giving special name to each resulting inverse function:
$\mathrm{ArcNem}_q$, $\mathrm{ArkNem}_q$, $\mathrm{ArqNem}_q$. The third of them, $\mathrm{ArqNem}_q$, happened to be satisfactory; so, I use it as
$\mathrm{Nem}^{-1}$.
The cut lines of function $\mathrm{ArqNem}_q$ go from $-\infty$ to zero, and then to each of the branch points, seen at the maps in the right column of Figure \ref{nema}. 
\vskip 6pt

For the positive values of the argument, curves in Fig. \ref{arqnemplo}
refer to any of these three functions, $\mathrm{ArcNem}_q(x)\!=\!\mathrm{ArkNem}_q(x)\!=\!\mathrm{ArqNem}_q(x)$ . Below I describe the only last one.
\vskip 6pt

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\begin{picture}(170,420)
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\put(120,284){\sx{1.2}{$q\!=\!3$}}
\put(122,263){\sx{1.2}{$q\!=\!4$}}
\put(122,250){\sx{1.2}{$q\!=\!5$}}
\put(112,200){\sx{1.2}{$q\!=\!10$}}
\put(104,164){\sx{1.2}{$q\!=\!20$}}
\put(94,145){\sx{1.2}{$q\!=\!30$}}
\put( 74,99){\sx{1.2}{$q\!=\!100$}}
\put( 49,52){\sx{1.2}{$q\!=\!1000$}}
\put( 36,32){\sx{1.2}{$q\!=\!10000$}}
\put(166,40){\rot{90}\sx{1.1}{\url{https://mizugadro.mydns.jp/t/index.php/File:Nembrant.jpg}}\ero}
\end{picture}}
\vskip -8pt
\caption{Parametric plot:
$x\!+\!\mathrm i y\!=\!\mathrm{NemBran}(q)$
\label{nembranfig} }
\vskip -144pt
\end{wrapfigure}
For handling of cuts of function $\mathrm{ArqNem}_q$, the branch points should be calculated. These branch points are shown in figure
\ref{nembranfig}. It refers to function $\mathrm{NemBran}$, that expresses the upper branch point of function $\mathrm{ArqNem}_q$ as function of $q$.
Below I describe, how to evaluate this function.
\vskip 6pt

Assume, some $q$ is given. We need to find solutions of equation $\mathrm{Nem}_q^{~\prime}(z)\!=\!0$.
This solution can be expressed in Mathematica language with code
 \begin{verbatim}
T[z_]=z+z^3+q z^4
s = Solve[T'[z]==0,z]
ReplaceAll[z,Extract[s,1]]
\end{verbatim}
The output indicates, how to program function NemBran, shown in figure \ref{nembranfig}:
\begin{verbatim}
z_type nembra(DB q){ z_type Q,v,V; 
  Q=q*q; 
  v=-1.-8.*Q+4.*sqrt(Q+4.*Q*Q); 
  V=pow(v,1./3.); 
   return (.25/q)*(-1.+1./V+V); }
\end{verbatim}

\begin{verbatim}
z_type NemBran(DB q){ z_type z,zz=z*z; 
  z=nembra(q); return z*(1.+zz*(1.+q*z)); }
\end{verbatim}
Here {\tt DB} denotes {\tt <double>}~ and~ {\tt z\_type} denotes {\tt complex<double>} ;
NemBran appears as combination of functions Nem and nembra. 
\vskip 6pt

Assume, some $q\!>\!0$ is given; let $x_0\!+\!\mathrm i y_0\!=\!\mathrm{NemBran}(q)$. 
For function $\mathrm{ArqNem}_q$, I draw the cut lines form $-\infty$ to zero, and from zero to point 
$(x_0,y_0)$ and to point $(x_0,-y_0)$. As the cut lines are specified, it is easy to program evaluation of
$\mathrm{ArqNem}_q$, picking up the corresponding branch of the solution. The C++ code is shown in Table \ref{T1}. One can extract the code also from the URL marked in figure \ref{nembranfig}.
The readers are invited to check numerically relations 

\be
\mathrm{Nem}_q\big(\mathrm{ArqNem}_q(z)\big)=z\\
\mathrm{ArcNem}_q\big(\mathrm{Nem}_q(z)\big)=z
\ee

and investigate the range of validity of each of these equation.

\begin{table}
\caption{C++ implementation of function $\mathrm{ArqNem}_q$. Values of 
$q$ and corresponding $x_0,y_0$ should be already calculated and stored in global variables Q, tr, ti 
\label{T1}} \normalsize \baselineskip 13.4pt
~
\vskip -44pt
~

\begin{verbatim}
z_type arnemU(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;
z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);
z_type R=-I*sqrt(-r);
z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=I*sqrt(-h);
z_type g=0.5/q2  - B + (.25+2.*q2)/(q3*H);
z_type G=I*sqrt(-g);
return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arnemD(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;
z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);
z_type R=I*sqrt(-r);
z_type s=27.*a + 3.*R;  z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=-I*sqrt(-h);
z_type g=0.5/q2  - B + (.25+2.*q2)/(q3*H);
z_type G=-I*sqrt(-g);
return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arnemR(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;
z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);
z_type R=sqrt(r); z_type s=27.*a + 3.*R;
z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=sqrt(h);
z_type g=0.5/q2  - B + (.25+2.*q2)/(q3*H);
z_type G=sqrt(g);
return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arnemL(z_type z){ DB q=Q; DB q2=q*q; DB q3=q2*q;
z_type a=q-z; z_type b=1.+4.*q*z; z_type r=81.*(a*a)+12.*(b*b*b);
z_type R=-sqrt(r);
z_type s=27.*a + 3.*R; z_type S=pow(s,1./3.);
z_type B=(0.26456684199469993*S)/q - (1.2599210498948732*b)/(q*S);
z_type h=0.25/q2 + B;
z_type H=sqrt(h);
z_type g=0.5/q2  - B + (.25+2.*q2)/(q3*H);
z_type G=sqrt(g);
return - 0.25/q - 0.5*H + 0.5*G ;}

z_type arqnem(z_type z){ DB x,y; x=Re(z);y=Im(z);
if( y>ti || (x<0 && y>=0)) return arnemU(z);
//if(y<0) return conj(arnemU(conj(z)));
if(y<-ti || (x<0 && y<=0)) return arnemD(z);
if(x*ti>fabs(y)*tr) return arnemR(z);
return arnemL(z);}
\end{verbatim}
\normalsize
\end{table}

\section{SuNem}
\baselineskip 18.8pt

\begin{wrapfigure}{r}{150px} \normalsize
\sx{.33}{
\begin{picture}(620,840)
\put(4,0){\ing{sunemplo4}}
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\put(224,700){\sx{3.2}{$7$}}
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\put(444,60){\rot{90}\sx{2.4}{\url{https://mizugadro.mydns.jp/t/index.php/File:Sunemplo4t.jpg}}\ero}
\end{picture}}
\vskip -9pt
\caption{$y\!=\!\mathrm{SuNem}_q(x)$ %versus $x$
\label{sunemplo}}
\end{wrapfigure}

Functions 
$\mathrm{Nem}_q$ and
$\mathrm{ArqNem}_q\!=\!\mathrm{Nem}_q^{-1}$
are defined, implemented, tested and described above; they can be used for construction of superfunction $\mathrm{SuNem}_q$, abelfunction  $\mathrm{AuNem}_q\!=\!\mathrm{SuNem}_q^{-1}$ and corresponding non-integer iterates of the Nemtsov function. I begin with function $\mathrm{SuNem}_q$. It is shown in figure \ref{sunemplo} for various values of $q$.
In this section, I describe, how is it constructed.
\vskip 9pt

For the Nemtsov function $\mathrm{Nem}_q$, the superfunction  is solution $F$ of the transfer equation
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F(z\!+\!1)=\mathrm{Nem}_q(F(z)) ~
\eL{Feq}
In analogy with approach of the previous chapter,
I look for solution $F$ with the certain asymptotic behaviour,
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F_q(z)\!=\!\frac{1}{\sqrt{-2 z}}\left(\! 1\!-\!\frac{q}{\sqrt{-2 z}}\! +\! O\Big( \frac{\ln(-z)}{z}\Big) \right)
~ ~ ~ ~ ~ ~  ~ ~
\iL{F}
\ee
In order to construct the computationally-efficient asymptotic approximation of superfunction $F_q$, define set of polynomials
\be
P_m(z)=\sum_{n=0}^{\mathrm{IntegerPart}[m/2]} a_{m,n} z^n
\eL{P}
where $a$ are constant coefficients. Then, I set
\be
F_{q,M}(z)=\varepsilon \left( 1- q \varepsilon + \sum_{m=2}^M P(\ln(-z)) \varepsilon^m \right)
\eL{FqM}
where 
\be
\varepsilon=\frac{1}{\sqrt{-2 z}}
\eL{epsilon}
I substitute the approximation \rf{FqM} into the transfer equation \rf{Feq}. The asymptotic analysis of the residual (id est, its asymptotic minimisation) determines coefficients $a$.
The asymptotic representation \rf{P},\rf{FqM},\rf{epsilon} should approximate superfunction $F$ at least for large negative values of the argument. 
\vskip 6pt

The original (and non-trivial) part of this research is guessing of representation \rf{P},\rf{FqM},\rf{epsilon}.
This representation is a little bit more complicated, that the similar representation 
\rf{susinasymf},
\rf{susinFM},
\rf{susina} for function SuSin, described in the precious section. 
\vskip 6pt

Once representation \rf{P},\rf{FqM},\rf{epsilon} is written out, 
the following construction is straightforward.
This analysis can be performed with the Mathematica code shown in Table \ref{T2}. 


\begin{table}[t]
\caption{Computation of coefficients $a$ in equation \rf{P} \iL{T2}}
\normalsize \baselineskip 13.4pt
\begin{verbatim}
T[z_] = z + z^3 + q z^4
P[m_, L_] := Sum[a[m, n] L^n, {n, 0, IntegerPart[m/2]}]
A[1, 0] = -q; A[1, 1] = 0;
a[2, 0] = 0; A[2, 0] = 0;
F[m_,z_]:=1/(-2z)^(1/2)(1-q/(-2z)^(1/2)+
Sum[P[n,Log[-z]]/(-2z)^(n/2),{n,2,m}])

m = 2;
s[m] =Numerator[Normal[Series[
     (T[F[m,-1/x^2]]-F[m,-1/x^2+1])2^((m+1)/2)/x^(m+2),{x,0,1}]]]
t[m] = Numerator[Coefficient[Normal[s[m]], x] ]
sub[m] = Extract[Solve[t[m] == 0, a[m, 1]], 1]
SUB[m] = Simplify[sub[m]]
f[m, z_] = ReplaceAll[F[m, z], SUB[m]]

m = 3
s[m] =Simplify[ReplaceAll[Series[
(T[F[m,-1/x^2]]-F[m,-1/x^2+1])2^((m+3)/2)/x^(m+3),{x,0,0}],SUB[m-1]]];
t[m] = ReplaceAll[Normal[s[m]], Log[x] -> L];
u[m] = Table[Coefficient[t[m] L, L^n]==0, {n,1,1+IntegerPart[m/2]}];
tab[m] = Table[a[m, n], {n, 0, IntegerPart[m/2]}];
sub[m] = Extract[Solve[u[m], tab[m]], 1]
SUB[m] = Join[SUB[m-1], sub[m]];

(* and so on for  m=4,  m=5, etc. *)
\end{verbatim}
\end{table}
\vskip 6pt

Coefficients $a$ are chosen in such a way, that 
\be
F_{q,M}(z)=F_q(z)+O\big(\varepsilon^{M+1}\big)
\ee
Tens of coefficients $a$ in equation \rf{P} can be computed in such a way. The first coefficients are: %\newcommand \ts {\textstyle}
\vskip 6pt
$a_{2,0} = 0$ ~~, ~~~~~~~  ~~~~~~       $ a_{2,1} = \frac{1}{4} (3 + 2 q^2)$
\vskip 6pt
$a_{3,0} = q + 3 q^3$ ~,~  ~~~~~~ $a_{3,1} = -(3 q)/2 - q^3$
\vskip 6pt
$a_{4, 0}\!=\!\frac{1}{8} (5 \!-\! 4 q^2 \!-\! 44 q^4) $,~ 
$a_{4, 1}\!=\!\frac{1}{8} (-9 \!-\! 12 q^2 \!-\! 4 q^4)$,~
$a_{4, 2}\!=\! \frac{3}{32} (9 \!+\! 12 q^2\! +\! 4 q^4)$ 
\vskip 6pt
$a_{5, 0}\!=\! \frac{1}{12} (-39 q \!-\! 104 q^3 \!-\! 4 q^5)$,~
$a_{5, 1}\!=\! \frac{7}{4} (3 q \!+\! 8 q^3 \!+\! 4 q^5)$,~
$a_{5, 2}\!=\! -\frac{9 q}{4} \!-\! 3 q^3 \!-\! q^5$
\vskip 6pt
Coefficients $a$ happen to be polynomials with respect to parameter $q$.
For the following numerical implementation of superfunction, they are expressed through the Horner rule.
The C++ implementation is loaded together with generators of figures of this chapter. 
\vskip 8pt

Assume given number $M$ of terms of sum in the asymptotic expansion \rf{FqM}. Then, the
superfunction $F_q$ can be defined as limit
\be
F_q(z)=\lim_{n\rightarrow \infty} \mathrm{Nem}_q^{\, n} \Big( F_{q,M}(z-n) \Big)
\eL{Flim}
I remind, the upper index after the name of the function indicates the number of its iterate.
The limit does not depend on the number $M$. However, for large $M$, the limit converges faster.
For $q$ of order of unity, and argument $z$ of order of unity, with $M\!=\!30$, it is sufficient to make $n\!=\!20$ iterates in order to approximate limit in equation \rf{Flim}
with 14 decimal digits. That greatly exceeds the precision, required to plot the camera-ready copies of all the figures presented. However, the extra digits help to reveal faults of the representation of function, if any mistake takes place.
\vskip 6pt

The transfer equation has translational invariance. If some $z \rightarrow F(z)$ is the solution, then, for a constant $C$, function $z \rightarrow F(z\!+\!C)$ is also solution, id est, also superfunction of the same transfer function. 
In order to make figures more beautiful it is convenient, that at zero, the superfunction has value unity.
For this reason, I define superfunction $\mathrm{SuNem}_q$ as superfunction $F$ with displaced argument:
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{SuNem}_q(z)=F_q(x_1\!+\!z)
\eL{SuNem}
where $x_1=x_1(q)$ is real solution of equation 
\be
F_q(x_1)=1
\ee
This provides condition
\be
\mathrm{SuNem}_q(0)=1
\ee
This condition is not so important to evaluate iterates of the Nemtsov function, but it helps to compare different superfunctions.
Many superfunctions, described in this Book, have value unity at zero.
\vskip 6pt

Definition \rf{SuNem} of function SuNem is used to generate the explicit plot in figure \ref{sunemplo} and also the
complex maps of $\mathrm{SuNem}_q$ for 
$q\!=\!0$,
$q\!=\!1$ and
$q\!=\!2$ in figure \ref{suma}.
These maps are symmetric with respect to reflection from the real axis, so, the only upper half of the complex plane is shown.% in each map.

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\begin{figure}[tp]
\vskip -10pt
{\url{https://mizugadro.mydns.jp/t/index.php/File:Sunem0ht.jpg}}

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\vskip 2pt

{\url{https://mizugadro.mydns.jp/t/index.php/File:Sunem1ht.jpg}}

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\vskip 2pt

{\url{https://mizugadro.mydns.jp/t/index.php/File:Sunem2ht.jpg}}

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\vskip -4pt
\caption{
$u\!+\!\mathrm i v=\mathrm{SuNem}_q(x\!+\!\mathrm i y)$, 
for $q\!=\!0,1,2$
\iL{suma}}
\end{figure}
\vskip 6pt

Maps in figure \ref{suma} demonstrate the asymptotic behaviour of function SuNem.
In the most of the complex plane, the function slowly decays to zero.
This decay is determined by the leading term of the asymptotic expansion \rf{FqM}.
However, this asymptotic expansion is not valid in vicinity of the origin of coordinate,
nor in vicinity of the positive part of the real axis. 
In the half-strips $x\!>\!2$, $1\!<\!|y|\!<\!2$, the maps show complicated, oscillating behaviour of the function.
For positive values of the argument, function $\mathrm{SuNem}_q$
shows fast growth. This growth is seen both, in the explicit plot at the right hand side picture of figure
\ref{sunemplo} and in the complex maps in figure \ref{suma}.
\vskip 8pt

For computation of iterates of the Nemtsov function $\mathrm{Nem}_q$,
the Abel function $\mathrm{AuNem}_q\!=\!\mathrm{SuNem}_q^{-1}$ is also required. Function $\mathrm{AuNem}_q$ is described in the next section.

\section{AuNem}
In this section, I construct abelfunction for the Nemtsov function $\mathrm{Nem}_q$ by \rf{nem}.
This abeldunction is inverse of the superfunction, id est, 
$\mathrm{AuNem}_q\!=\!\mathrm{SuNem}_q^{-1}$.
\vskip 8pt

First, consider inverse function of the superfunction $F$ by \rf{F}; let $G_{q}\!=\!G\!=\!F^{-1}$. It 
can be expanded as follows:
\be \nonumber
\mathrm{G}_{q,M}(z)&=&
-\frac{1}{2 z^2}+\frac{q}{z}
+\frac{1}{2} \left(2 q^2\!+\!3\right) \log (z)
+\frac{q^2}{2}+\frac{1}{4} \left(2 q^2\!+\!3\right) \log (2)
\\ %\nonumber
&& 
+ ~ \sum _{n=1}^M c_n z^n  
\iL{auneqm}
\\
G_{q}(z)&=&\mathrm{G}_{q,M}(z) +O(z^{M+1})
\label{auneqa}
\eL{G}
This form can be obtained, inverting expansions \rf{F}, \rf{FqM} for superfunction $F_q$.
\vskip 8pt

Coefficients $c$ depend on $q$; these coefficients can be computed either with asymptotic analysis of equation
\be
G_{q,M}(F_{q,M}(x))=z
\ee
or from the Abel equation
\be
G_q(\mathrm{Nem}_q(z))=G_q(z)+1
\ee
Coefficients $c$ in equation \rf{auneqm} can be calculated in Mathematica with the code shown in Table
\ref{Gmat}.
\baselineskip 18pt

\begin{table}[!ht]
\caption{Mathematica code for calculation of coefficients $c$ in equation \rf{auneqm} \label{Gmat} }
\normalsize
\baselineskip 13.6pt
\begin{verbatim}
T[z_] = z + z^3 + q z^4
P[m_,L_]:= Sum[a[m, n] L^n, {n, 0, IntegerPart[m/2]}]
F[m_,z_]:=1/(-2z)^(1/2)(1-q/(-2z)^(1/2)+
                         Sum[P[n,Log[-z]]/(-2z)^(n/2),{n,2,m}])
G[m_,x_]:=-1/(2x^2)+q/x+q^2/2+1/4(3+2q^2)Log[2]+1/2 (3+2q^2)Log[x]+
                                                Sum[c[n]x^n,{n,1,m}]
Series[ReplaceAll[F[3,h+G[3, z]], a[2,1]-> 1/4 (3+2 q^2)], {z,0,4}] 
(*The line above is just test *)
m=1;
sg[m]=Coefficient[Series[G[m+3,T[z]]-G[m+3,z]-1,{z,0,3}], z^(m+2)]
st[m]=Solve[sg[m] == 0, c[m]]
su[m]=Extract[st[m], 1]
SU[m]= su[m];

m= 2; 
sf[m]=Series[ ReplaceAll[G[m+3,T[z]]-G[m+3,z]-1,SU[m-1]],{z,0,m+2}]
sg[m]=Simplify[Coefficient[sf[m] 2^m, z^(m+2)]]
st[m]= Solve[sg[m] == 0, c[m]]
SU[m]= Join[SU[m - 1], su[m]]

m = 3;
sf[m]=Series[ ReplaceAll[G[m+3,T[z]]-G[m+3,z]-1,SU[m-1]],{z,0,m+2}]
sg[m]=Simplify[Coefficient[sf[m] 2^m, z^(m+2)]]
st[m]= Solve[sg[m] == 0, c[m]]
su[m]= Extract[st[m], 1]
SU[m]= Join[SU[m-1], su[m]]

(*and so on for m=4, m-5, etc... *)
\end{verbatim}
\end{table}

\vskip 6pt
For an integer $M\!>\!0$, the abelfunction
$G_q$ can be evaluated through the asymptotic approximation $G_{q,M}$ as the limit
\be \!\!\!\!\!\!\!\!\!\!\!\!\!
G_q(z)=
\lim_{n\rightarrow \infty} \Big(\mathrm{G}_{q,M}\big(\mathrm{ArqNem}_q^{~n}(z)\big)+n\Big)
\eL{AuNemLim}
Then, the function AuNem is expressed through function $G$ with addition of the constant,
\be
\mathrm{AuNem}_q(z)=G_q(z)-G_q(1)
\ee
in such a way, that $\mathrm{AuNem}(1)\!=\!0$.
The readers are invited to check numerically the ranges of validity of relation
\be
\mathrm{SuNem}_q\big(\mathrm{AuNem}_q(z)\big)=z \\
\mathrm{AuNem}_q\big(\mathrm{SuNem}_q(z)\big)=z
\ee
and also estimate the residuals at the substitution of the numerical implementations into these relations.
\vskip 6pt

\begin{figure}
\parindent 0pt
\hskip 22pt
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\hskip 22pt \url{https://mizugadro.mydns.jp/t/index.php/File:Aunem1ht.jpg}\\
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\caption{$u\!+\!\mathrm i v=\mathrm{AuNem}_q(x\!+\!\mathrm i y)$ for 
$q\!=\!0$,
$q\!=\!1$,
$q\!=\!2$
\iL{aunema}}
\end{figure}
Complex maps $u\!+\!\mathrm i v=\mathrm{AuNem}_q(x\!+\!\mathrm i y)$ for 
$q\!=\!0$,
$q\!=\!1$ and
$q\!=\!2$ are shown in the $x,y$ plane in Fig. \ref{aunema}.
\vskip 8pt

The reader is invited to compare the cut lines in figure \ref{aunema} with cuts in maps at the right hand side column of figure {suma}.
The cut lines of function $\mathrm{AuNem}_q$ are the same, as those of function $\mathrm{ArqNem}_q$;
because it is the first function to evaluate while computing through the asymptotic representation $G_{q,M}$ by equation \rf{AuNemLim}. 
The branch points are determined by function $\mathrm{NemBran}(q)$ shown in figure \ref{nembranfig}.
\vskip 8pt

Lines $v\!=\!\mathrm{const}$ in maps of figure \ref{aunema} approach the cut lines in pretty specific way, that corresponds to decrease of $u$, which represents the real part of the function. This means, that, if one goes along any line $v\!=\!\mathrm{const}$, increasing $u$, one approaches to the origin of coordinates, where the asymptotic representation is accurate, without to cross the cut lines.
This happens due to the specific choice of the inverse function $\mathrm{ArqNem}_q$,
with this goal, the cut lines of function  $\mathrm{ArqNem}_q$ are chosen in the function ArqNem.
(The default choice of the inverse function, provided by the routine ``Solve'' in Mathematica, does not provide this.)
\vskip 9pt

In such a way, the Abel function AuNem for the Nemtsov function is constructed and implemented.
With functions 
$\mathrm{SuNem}_q$ and 
$\mathrm{AuNem}_q$, the iterates
of function $\mathrm{Nem}_q$ 
can be calculated. 
They are shown in figures \ref{iteraplo}, \ref{niteramap}
and 
described in the next section.

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\vskip -2pt
\caption{
$y\!=\!\mathrm{Nem}_q^{\,n}(x)$ at $q\!=\!0$, left, and $q\!=\!2$, right, for various  $n$ by \rf{Nqn} 
 \label{iteraplo} }
\vskip -2pt
\end{figure}

\section{Iterates of the Nemtsov function}
With functions  $\mathrm{Nem}_q$ and $\mathrm{SuNem}_q$ from the previous sections,
the iterates of the Nemtsov function can be defined, 
\be
\mathrm{Nem}_q^{\,n}(z)=\mathrm{SuNem}_q\Big( n + \mathrm{AuNem}_q(z) \Big)
\eL{Nqn}

Figure \ref{iteraplo} shows the explicit plot $y\!=\!\mathrm{Nem}_q(x)$ for 
$q\!\rightarrow\!0$ in the left hand side picture and for $q\!=\!2$ in the right hand side picture, for various values of the number $n$ of iterate. The integer values of $n$ 
correspond to the thick lines.
\vskip 6pt
For $q\!=\!1$, complex maps of iterates $\mathrm{Nem}_1^{\,n}$ are shown in figure \ref{niteramap}. The figure shows, how, at the increase of number $n$ of iterate from $-1$ to $1$,
function $\mathrm{ArqNem}_1$ gradually changes to the identity function (with rectangular grid as the complex map) and then to the Nemtsov function $\mathrm{Nem}_1$.
\vskip 8pt

Iterates in figure \ref{iteraplo} look similar to iterates of other quickly growing holomorphic functions
\cite{analuxp,vladie,sqrt2,factorial,sutra}.
In particular, at $n\!\approx\! 0$, the iterate $\mathrm{Nem}_q^{\,n}$ looks similar to identical function; at $n\!=\!1$, it is just Nemtsov function $\mathrm{Nem}_q$, and at $n\!=\!-1$, it is the inverse function, id est, $\mathrm{ArqNem}_q$. 

Iterates of a growing real-holomorphic function are also real-holomorphic; the complex maps are symmetric with respect to reflection from the real axis, so, the only upper half of the complex plane is shown shown in figure \ref{iteramap}. The left column shows maps for the positive iterates; the number $n$ varies from $0.6$ at the top map with step $-0.1$ to $-0.1$ at the bottom map.
In the similar way, the right hand side column represents maps for $n$ from $-0.6$ at the top to $-0.1$ at the bottom. Only case with $q\!=\!1$ is presented, but one can download the generator of the figure and plot similar maps for other values of $q$, and, of course, other values of number $n$ of iterate; this number can be even complex.
\vskip 8pt

Iterates by \ref{Nqn} shown in figures  \ref{iteraplo},  \ref{iteramap} provide the smooth (holomorphis) transition from the Nemtsov function $\mathrm{Nem}_q$ to the identity function and then to the inverse function ArqNem. Iterates have the group property,
\be
T^{m+n}(z)=T^{m}(T^n(z))
\eL{Tmn}
This ratio holds only for certain range of values of parameters, that includes the positive part of the real axis for $z$. 
To keep the Book of reasonable thickness, I skip this analysis and invite Readers to do this as an exercise.

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\caption{
$u\!+\!\mathrm i v\!=\!\mathrm{Nem}_1^{\,n}(x\!+\!\mathrm i y)$  for
$n=0.6$ .. $0.1$, left, and for  $n=-0.6$ .. $-0.1$, right, by \rf{Nqn}
\iL{iteramap}
\label{niteramap}
 }
\end{figure}

\section{End of exotic iteration}
\baselineskip 18.8pt

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\caption{\baselineskip 9px
B.Nemtsov \label{boris}}
\vskip -22pt
\end{wrapfigure}

The special 4th order polynomial \rf{nem} is considered in this section. I call it Nemtsov function after Boris Nemtsov, Fig. \ref{boris}; the need to denote this function coincided with the tragic event. This polynomial is treated as transfer function: the inverse function ArqNem, the superfunction SuNem and the abelfunction AuNem are constructed. 
 %$\mathrm{AuNem}_q$ in the whole complex plane.
\vskip 8pt
 
Construction of function ArqNem happened to be non-trivial; so, its map is shown in the right hand side column of figure \ref{nema} for various values of parameter. Choice of the cut lines of this function is important. The readers are invited to try to construct abelfunction with other choice of the cut lines and plot the complex maps of the result. And compare the resulting complex maps to those in figure \ref{aunema}. And the same for the iterates in figure \ref{iteramap}.
\vskip 8pt
 
Function Nem is my last attempt to construct a difficult-to-iterate growing real-holomomorphic function with real fixed point.
The fixed point is chosen at zero, because the update to the more general case is straightforward, it is specified in the last row of table
\ref{TableOfSuperfunctions}.
\vskip 8pt

For the real-holomorphic growing transfer function with real fixed point, the suprfunction can be constructed with regular iteration
considered in chapter 6, if the derivative at fixed point is not unity, or with exotic iteration, if this derivative is unity.
Then, one can guess the heuristic ``solution'' with correct asymptotic behaviour of the superfunction, using analogy with the differential equation, discussed in the previous chapter; the same analogy works for the Nemtsov function too. This leads to iterates that I call ``exotic''.
The exotic iterates lead to pretty regular and real-holomorphic superfunctions, abelfunctions and correspondingly regular non-integer iterates of the transfer function.
\vskip 8pt

In the next chapter, I consider, in some sense, even more exotic case, when the transfer function has no real fixed point.
The example of such a function without real fixed point is just natural exponent, $T=\exp$.
Historically, namely this transfer function had been considered and iterated (it can be iterated any real or even complex number of times) 
\cite{analuxp}.


\chapter{~ Natural tetration ~ tet}
\baselineskip 20pt

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\caption{\large \baselineskip 11pt
$y\!=\!\mathrm{tet}(x)\,$,
$\,y\!=\!\exp(x)\,$, 
~and ~~$\,y\!=\!10\big(\mathrm{tet}(x)-(x\!+\!1)\big)~$ \iL{TetPlot}}
\vskip 18pt
\end{wrapfigure}

Here, I consider the exponential transfer function, $T\!=\!\exp$.
For this transfer function, the transfer equation can be written as follows:
\be
f(z\!+\!1)=\exp(f(z)) ~ ~ ~
\label{transferexp}
\ee
In order to narrow the set of solutions, the additional condition is assumed:
\be
f(0)=1
\label{f0172}
\ee
In order to provide the uniqueness, in addition, I require, that the solution $f(z)$ is holomorphic
at the whole complex plane except $z\!\le\!-2$, 
and also limited at least in the strip $\Re(z)\!\le\! 1$.
I refer the solution $f$ as ``tetration'' (or ``natural tetration'') and denote it with symbol $\mathrm{tet}$.
In this chapter I tell, how this function is constructed; I use the main formulas and pictures from publications
\cite{analuxp,vladie}.
\vskip 2mm

Colleagues ask questions not only about superfunctions and way of the evaluation, but also about guessing of properties of these functions. In particular, about the natural tetration:
``How did you guess?''
\vskip 2mm

The questions about guessing are important not only for the history,
but also for the colleagues, who want to use the similar way for other transfer functions, superfunctions and, perhaps, even to some more complicated objects. So, I consider these questions seriously. Especially, this apply to the natural exponent and the natural tetration, as the natural tetration is first non-trivial superfunction, for which the efficient algorithms of evaluation had been suggested and described \cite{analuxp,vladie}. 
\vskip 2mm

In this chapter, I provide not only the formulas and pictures for the natural exponental and the natural tetration, but also explanations, why namely this tetration should be considered and recognised as the most ``true'' and the ``most natural'' among various possible superfunctions; and why any researcher, following the same idea, should come to the same tetration.
\vskip 2mm

\section{Exponent}
Before to deal with solution  $f$ of equations \rf{transferexp}, \rf{f0172},
it worth to remind properties of exponent.
For the real argument, graphic of exponent is shown in figure  \ref{TetPlot} with thin line.
Complex map of the transfer function  $T=\exp$ is shown in figure \ref{expMap}.
\vskip 2mm

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\vskip 24pt
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\vskip 19pt
\end{figure}

I hope to be not condemned for drawing so elementary functions as exponent. I did not do it in the previous section about sin; but for some elementary functions I provide the maps. The Book should allow the understanding, even if it happens to be in hand of a pure experimentalist. With the detailed descriptions of the elementary things, I hope, even 
Akira Shirakawa and Yulya Kuznetsova can understand at least the main idea of the Book. 
The Book must allow the reading even by the least-educated academician
\cite{stru}
\footnote{\url{https://royallib.com/read/Strugatsky_Arkady/Tale_of_the_Troika.html}  Tale of the Troika by Arkady and Boris Strugatsky. PROLOG.
{\em "..Our slogan is ‘elevators for everyone.’ No matter who. The elevator must be able to withstand the entrance of the least-educated academician.”}}.
\vskip 2mm

The exponent can be considered as superfunction of function ``multiplication by constant number e''; 
$\mathrm e\!=\!\exp(1)\!\approx\!2.71828182846\,$. In such a way, exp is superfunction for the transfer function $T$ by
\be
T(z)=\mathrm e \, z
\eL{Tez}

I repeat the formula from the school course of algebra:
\be
f(z\!+\!1)=\mathrm e \, f(z)
\eL{expz1ez}

The solution of this equation can be constructed with regular iteration, in vicinity of the fixed point $L\!=\!0$ of the transfer function $T$ by
\rf{Tez}.
The Reader is invited to make this exercise and check, that the primary expansion stops at the first term, giving the exact solution
$f\!=\!\exp$ at the first iteration.
\vskip 2mm

For the transfer function $z\!\mapsto \mathrm e \,z$, I write also the Abel equation:
\be
g(\mathrm e \, z)=1+ g(z)
\eL{abelog}
Equation  \rf{abelog} is considered by Henryk Trappmann
\cite{log}. 
The Reader can guess, that Henryk got the natural logarithm as the solution, id est, $g\!=\!\ln$. 
Readers are invited to think, what additional requirements should be associated with equation
\rf{abelog}, in addition to equation 
\be g(1)=0\ee
in order to provide the uniqueness of the solution $g\!=\!\ln$.
\vskip 2mm

Explicit plot of exp is shown in figure \ref{ExpQ2ePlot}. 
(The same dependence is shown with thin curve in figure \ref{TetPlot}.)
It is worth comparing the graphic of the natural exponent to that of the exponent to base $b\!=\!\sqrt{2}$, which crosses the straight line $y\!=\!x$. 
This line is also shown in figure
\ref{ExpQ2ePlot}.
The graphic of the natural exponent
$y\!=\!\exp(x)$  
does not cross the line 
$y\!=\!x$.
The natural exponent has no real fixed point.
\vskip 2mm

Fixed point  $L$ of exponential and logarithm to base $b\!=\!\exp(a)$ is solution of equation $L\!=\!\log_b(L)$. 
This solution can be expressed through the Tania function, considered above in chapter 4:

\be\!\!\!\!\!\!
L\!=\!\mathrm{filog}(a)\!=\!
\frac{\mathrm{Tania}(\ln(a)\!-\!1\!-\! \pi \mathrm i) }{-a}\!=\!
\frac{\mathrm{WrightOmega}(\ln(a)\!-\! \pi \mathrm i) }{-a} ~
\label{Lexp}
\ee
The second equality in formula \rf{Lexp} can be considered as definition of the new function filog. This function is described also in TORI, \\
\url{https://mizugadro.mydns.jp/t/index.php/Filog}.\\
Function filog is considered in details below, in chapter 18. Here, we need this function only for the  single value of the argument, namely, $L=\mathrm{filog}(1)$.

As function Tania is already described, it should be considered as special function. In this sense, quantity $L$ should is exact:
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
L\!\!&=&- \mathrm{Tania}(-1\!-\!\pi\mathrm i) \nonumber \\
&\approx& 
  0.3181315052047641353%12654
+1.3372357014306894089%0116
\, \mathrm i
\label{expL}
\ee
The rough approximation (with two significant figures) for $L$  by \rf{expL} 
 can be found even from figure  \ref{figTaniaMap}, counting isolines with finger.
\vskip 2mm

Function Tania is used in this Book already twice: first, in chapter 2, as realistic superfunction for the transfer function Doya
(that describes increase of the intensity of light in a laser amplifier with simple model of continuously pumped of active medium) and, second time, here, as exact representation for the fixed point of exponent. This is the same function. Recycling, reusing of the results is base of any science, and the physical mathematics (see the Introduction) is not an exception.
\vskip 2mm

Exponent is real-holomorphic function, 
$\exp(z^*)=\exp(z)^*$;  
hence
$\,L^*\!\approx\! 0.1\!-\!1.3\,\mathrm i\,$
is also the fixed point. In principle, each of these fixed points can be used for the regular iteration, considered in chapter 6.
However, such iteration does not lead to the real-holomorphic superfunction. I wanted to suggest a way of evaluation of the real-holomorphic tetration, that could be used as the definition. In order to specify it, I assume, that there exist some special superfunction of the natural exponent, and this superfunction is characterised with specific 
behaviour. As the real part of the argument goes to $-\infty$, the superfunction approaches $L$ in the upper part of the complex half plane, and $L^*$ in the loser part. At the beginning, this is nothing more, but just guess.
In the next section, this guess is used 
to construct both, the definition and the way of evaluation. 

\section{Heuristic tetration}
\baselineskip 18pt

Since publication of the first article about real-holomorphic tetration, the colleague ask me similar questions:

``Why did you interested in holomorphic tetration?"

``How did you guess the asymptotic behaviour to tetration at  $\pm \mathrm i \infty$?''

``How did you guess the initial approximation $\mathrm{fit}_3$?'' \cite{guess}. 

In order not to be like Rip van Winkle \footnote{
\url{https://classiclit.about.com/library/bl-etexts/wirving/bl-wirving-rip.htm}
Rip Van Winkle by Washington Irving. (1783-1859)
}, 
revealing new and new details, %and motivations, that led me to the tetration, described in this section,
here I mention the only one of motivations, that is related to physics.
\vskip 6pt

Initially, I wanted to use some fast-growing function in order to represent the factorial of the number of particles in the Bose-Einstein condensate, this factorial appears at the attempt to write-out the first approximation for the normalized multi-particle wave function. 
The fast-growing function is described in the article by Hooshmand \cite{hooshmand}, but it happened to be not suitable for the asymptotic analysis, because it is not holomorphic.
The corresponding extension
\be\!\!\!\!\!\!\!\!\!\!
f(z)=\mathrm{uxp}(z)=\!\left\{\!\!\!
\begin{array}{ccccc cc}
\ln\!\big({\rm uxp}(z\!+\!1)\big)~&{\rm at}&    \!&\!   \!&\! \Re(z) \!&\! \le \!&\! -1 \\
                     z  +  1      ~&{\rm at}& -1 \!&\! < \!&\! \Re(z) \!&\! \le \!&\!  0 \\
\exp\!\big({\rm uxp}(z\!-\!1)\big)~&{\rm at}&  0 \!&\! < \!&\! \Re(z) \!& 
\end{array}
\right.
\label{uxp}
\ee
has many cut lines; they divide the complex plane to almost separated strips.
Complex map of function uxp by \rf{uxp} is shown in the upper picture of figure \ref{analuxpmap} with lines of constant log amplitude $u$ and lines of constant phase $v$,
\be
\exp(u\!+\!\mathrm i v) = f(x\!+\!\mathrm i y)
\ee
This representation is different from that, used for the most of complex maps in this Book;
usually, the lines of constant real part and those of constant imaginary part are drawn.
While I explain, how did I get the holomorphic tetration, I represent maps in the same form, as they appear in the original paper \cite{analuxp}.
\vskip 6pt

\begin{figure}
\normalsize
\hskip 8pt
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\vskip 8pt

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\vskip 8pt

\hskip 8pt
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\vskip 8pt

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\url{https://mizugadro.mydns.jp/t/index.php/File:Analuxp01u400.jpg}
\caption{\large $\exp(u\!+\!\mathrm i v)=f(x\!+\!\mathrm i y)$  
for the following functions:
$f\!=\!\mathrm{uxp}$, (a);
$f\!=\!\mathrm{Fit}_3$, (b); 
$f\!=\!\mathrm{Fit}_6$, (c); 
$f\!=\!\mathrm{tet}$, (d) % показаны уровнами логамплитуды и фазы: 
\label{analuxp01} 
\label{analuxp}
\iL{analuxpmap} }
\end{figure}

As I already mention above, the vertical cuts of the range of holomorphism of function uxp divide the complex plane to almost independent strips seen in the top map in figure \ref{analuxpmap}. These strips raise the question: 
Is it possible to suggest a ``more holomorphic'' (id est, with less cuts) solution of the transfer equation \rf{transferexp}? 
Or no solution $f$ of the equation  \rf{transferexp} may have wide range of holomorphism?
\vskip 6pt

At the first look into results by Hooshand \cite{hooshmand}, construction of the real-holomorphic solution seemed to be impossible.
On the other hand, the initial assumption used there, about monotonous derivative (for the real argument) of the superfunction of exp, looks doubtful.
I tried to construct some alternative proof, that the cuts are unavoidable, without to use the strange assumption.
I begun to investigate the case, assuming existence to the holomorphic solution. I expected to get some contradiction, and to use the contradiction for the proof. The holomorphic solution had been constructed, and 
no contradiction had been detected \cite{analuxp}; so, I had to accept the existence.
\vskip 6pt

Historically, the construction of this solution begun with approximations.
I had considered several real-holomorphic elementary functions, that have logarithmic singularity at $-2$ and take the same values, as tetration, at few integer values of the argument. One of them (that happened to be better than some others) is
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{fit}_2(z) \!=\! 
\ln(2\!+\!z)& \nonumber \\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+ (1\!+\!z)\left(
        1 + 
        \frac{z}{2}\exp\!\Big((z\!-\!1)s_2(z)\Big)
        			\Big(\!\mathrm e\! -2\! +\! \ln\frac{4}{3}	\Big)
		- \ln 2 
        \right)
        \label{fit2}
        \ee
where
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
s_2(z) = \exp\!\Big(\exp(z-2.51)\Big)-0.6+0.08(z\!+\!1)
\eL{fit2s}
Constants in the expression \rf{fit2s} are chosen in order to minimise the residual at the substitution
$f\!=\!\mathrm{fit}_2$ 
into the transfer equation \rf{transferexp}.
This approximation could be improved, comparing \rf{fit2s} with the precise approximation through the Cauchy integral, considered below. However, at the heuristic search for the rough approximations, the representation through the Cauchy Integral had not yet been written; so, choosing the approximation, I had to use the residual as the criterion.
\vskip 6pt

After construction of function $\mathrm{fit}_2$, 
it happened, that the linear combination of functions
$z\!\mapsto\!\mathrm{fit}_2(z)$ and  $z\!\mapsto\!\ln(\mathrm{fit}_2(z\!+\!1))$ 
gives the residual even smaller; 
in such a way, the approximation $\mathrm{fit}_3$ appeared:
\be
\mathrm{fit}_3(z) \!=\! 0.6\!~\mathrm{fit}_{2}(z)+0.4\!~\ln\big( \mathrm{fit}_{2}(z+1)\big)
\ee
The range of approximation of tetration can be extended. Let
\be \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\! \!\!\!\!
\mathrm{Fit}_3(z) \!=\!  \left\{\!\!
\begin{array}{ccccc cc}
\ln\!\big({\rm Fit}_{3}(z\!+\!1)\big)~&{\rm at}&~   \!&\!   \!&\! \Re(z) \!&\! \le \!&\! -1 \\
                \mathrm{fit}_{3}(z)           ~&{\rm at}& -1 \!&\! < \!&\! \Re(z) \!&\! \le \!&\!  0 \\
\exp\!\big({\rm Fit}_{3}(z\!-\!1)\big)~&{\rm at}&~ 0 \!&\! < \!&\! \Re(z) \!& 
\end{array}
\right.
\eL{Fit3}
Logampliture and phase of this function are show in figure
\ref{analuxp01}b. 
\vskip 2mm

For comparison, two more maps are shown in figure \ref{analuxp01}, they are numbered as c and d.
There represent the asymptotic approximation, %described in the next section,
$f=\mathrm{Fit}_6$
and tetration
$f=\mathrm{tet}$, 
described below in section 4; namely this tetration is goal of this chapter. 
\vskip 6pt

The asymptotic approximation
\be\!\!\!\!\!\!\!\!
\mathrm{Fit}_6(z) = \left\{
\begin{array}{c}
L+\exp(k(z\!+\!r))
 ~, ~ \Re(z)<-8
\\
\exp\Big(\mathrm{fit}_6(z\!-\!1)\Big)
~,~  \Re(z)\ge -8
\end{array}
\right.
\eL{Fit6}
is good at large values of imaginary part of the argument.
For natural tetration, the increment 
$k\!=\!L$. 
This looks as just coincidence. However, everyone can check it with the asymptotic analysis, substituting 
the primary expression of $\mathrm{fit6}$ into the transfer equation \rf{transferexp}.
Value of constant
$r\!\approx\! 1.075820830781 \!-\! 0.9466419207254 \,\mathrm i$ 
appears as adjusting parameter of this approximation.
On the other hand, it seems to be approximation of the important mathematical constant.
I would call this $r$ with name ``The Kneser constant''. This is one of constants, required for the expansion of iterates of exponent discussed in 
\cite{kneser} and used for the approximation of tetration \cite{vladie}.
\vskip 6pt

Function $\mathrm{Fit}_6(z)$ approximates tetration $\mathrm{tet}(z)$ at $\Im(z)>0.4$; 
function $\mathrm{Fit}_6(z^*)^*$ approximates $\mathrm{tet}(z)$ at $\Im(z)< - 0.4$;
combination of these functions is shown in map  ``c'' in figure \ref{analuxp01}.
In vicinity of the real axis, 
roughly,
in the strip $|\Im(z)|<0.4$, both these functions $\mathrm{Fit}_6(z)$ and $\mathrm{Fit}_6(z^*)^*$ look ugly,
and this strip in the map is left white.
\vskip 6pt

Approximations $f\!=\!\mathrm{Fit}_3$ and  $f\!=\!\mathrm{Fit}_6$ by \rf{Fit3} \rf{Fit6} 
are already sufficient to plot the complex maps and explicit plots of tertration; together, they provide of order of 3 decimal digits in the range of maps shown in figure \ref{analuxpmap}. The last map "d" in figure  \ref{analuxpmap} visually looks as superposition of the maps "b" and "c" above; this gave the general view of tetration that had to be constructed.
\vskip 6pt

The approximations above (even $\mathrm{Fit}_3$) allow to guess the asymptotic behaviour of tetration.
It should approach the fixed points $L$ or $L^*$ of logarithm, while the imaginary part of the argument approaches the plus or minus infinity. These values are indicated in maps ``c'' and ``d'' of figure  \ref{analuxpmap}.
\vskip 6pt

In such a way, this section explains, how did I guess, which the asymptotic behaviour should the tetration have.
Postulating this behaviour, one can construct the algorithm for evaluation of tetration with any required precision.
The postulated properties of tetration 
are collected in the next section.

\section{Properties of tetration}

Following recommendations by colleagues, friends and relatives, this content of this Book gradually goes from the simple examples to the more general formulas. In order to follow this way, in this chapter, I define only the natural tetration.
This section continues following article
\cite{analuxp} of year 2009.
. %Здесь предлагается определение тетрации и алгоритм для её вычисления.
\vskip 2mm

Since the two upper maps in figure \ref{analuxp01} were plotted, the main properties of this function are clear.
I postulate them below.
\vskip 2mm


Solution $F$ of the transfer equation \rf{transferexp} with additional condition $F(0)\!=\!0$ 
is called natural tetration, or simply tetration
$\mathrm{tet}$,
if the following conditions are satisfied:
\vskip 2mm

{\bf Т1.} 
Function $F(z)$ is real-holomorphic in the whole complex plane except the halfline $z\le -2$. Id est, $F(z^*)\!=\!F(x)^*$.
At $z\!=\!-2$, function $F(z)$ 
has logarithmic singularity, id est, the branch point.
\vskip 2mm

{\bf Т2.}  
Function $F(z)$ is bounded in the strip $|\Re(z)|\le 1$.
\vskip 2mm

{\bf T3}. 
Function $F(x)$ asymptotically approaches the fixed point $L$ in the upper half plane: for any real $x$, the relation below holds:
\be
\lim_{y \rightarrow + \infty} f(z+\mathrm i y)=L
~ ~ ~ ~ \eL{T715}
In addition, for positive $y$, the relation below holds:
\be
\lim_{x \rightarrow - \infty} f(z+\mathrm i y)=L
~ ~ ~ ~ \eL{T716}
\vskip 2mm

{\bf T4.} 
In the strip $~-1\le\Re(z)\le 2 ~$, the following condition holds:
\be
\arg(F(z))<2 %\pi
\eL{argF}
\vskip 2mm

Conditions T1-T4 above are a little bit redundant. The following development of the formalism of superfunctions is expected to indicate, which of these properties should be kept as definition of tetration, and which should appear as theorems, following from the shortened definition.
\vskip 2mm

From postulates \rf{T715}, \rf{T716} and real holomorphism  $f(z^*)=f(z)^*$, it follows, that
\be
\lim_{y \rightarrow - \infty} f(z+\mathrm i y)=L^*
~ ~ ~ ~ \eL{T717}
and for negative $y$, the relation below holds
\be
\lim_{x \rightarrow - \infty} f(z+\mathrm i y)=L^*
~ ~ ~ ~ \eL{T718}
These conditions are used in the next section for construction and evaluation of tetration through the Cauchy integral.

\section{Cauchy integral \label{section:IntegralCauchy}}
\baselineskip 18pt

For holomorphic function $F$, the Cauchy formula takes place \cite{Cauchy}:
\be 
F(z) = \frac{1}{2\pi \rmi} \oint_{\Omega} ~ \frac{F(t)}{t\!-\!z}~  \mathrm d t
\eL{Cauchy}
where contour $\Omega$ belongs to the simply connected range of holomorphism of function $F$ and only once passes around point $z$ in the counter clock wise direction.
With equation  \rf{Cauchy}, the transfer equation
\rf{transferexp}
leads to the integral equation for the values of superfunction along the imaginary axis
\cite{analuxp}. The description is repeated below.
\vskip 6pt

Engineer, physicist or mathematician, using formula \rf{Cauchy}, has certain freedom in choice of the contour of integration.
The researcher acts in a way, similar to that of an Engineer, who makes a project of the loop railroad for the rocky semi-island.
\vskip 3pt
The smart engineer takes into account the locations of ports, cities, farms, industries, in order to help the people to reach the places of their destination. Also, the engineer tries to avoid swamps, steep slops, narrow curvy canyons, to make the railroad fast, cheap and safe.
\vskip 3pt

Vainglorious tyrant, dictator, already famous in sports, war, art, archeology and ornithology, who wants to show himself also as a powerful all-mighty engineer,
may draw a rectangle on the map, and promote it as a project of the trace of the railway. Such a `` project '' will require a lot of bridges, ramps, excavations and tunnels, makes him famous also as vain waster of the state budget and may bring him to the situation ``no money'' \footnote{
\sx{.8}{\url{https://weirdrussia.com/2016/05/28/meme-medvedev-says-we-have-no-money-but-you-hang-in-there/}}
Medvedev Says “We have no money, but you hang in there” (2016).\\ %May 23rd, Prime Minister Dmitry Medvedev got a reality check, during a visit to Crimea. ..\
\url{https://www.bbc.com/news/blogs-trending-36482124} Russian PM: 'No money for pensions, but have a good day!' 2016.06.09\\
\sx{.82}{\url{https://www.cnbc.com/2016/06/09/there-is-no-money-left-bye-russian-pm-causes-social-media-storm.html}}
Holly Ellyatt. 'There is no money left, bye!': Russian PM causes social media storm. 
.. "no money left" in Russia's budget.. "There just isn't any money now. .."}.
\vskip 3pt

Sorry, in the choice of the contour of integration, described in  \cite{analuxp}, 
I look like as a tyrant, rather than as a smart engineer: I choose the contour of integration in the shape of rectangle.
The only excuse is, that the this contour leads to efficient way of evaluation of tetration. % and can be used as its definition.
\vskip 2mm

Let $F$ be real-holomorphic of equation \rf{transferexp}; %I repeat it one more:  $F(z\!+\!1)=\mathrm e^{F(z)}$.
\JP{ \be \lim_{y\rightarrow \infty} F(x+\mathrm i y)=L \ee
Тогда 
\be \lim_{y\rightarrow - \infty} F(x+\mathrm i y)=L^* \ee    }%JP

Let  $A$ be real positive number, so big, that $F(\mathrm i A) \!\approx\! L$ %and, therefore,  $F(\mathrm - i A) \!\approx\! L^*$ 

Let the range of hlomorphizm of function $F(z)$ includes the domain $-1\! \le\! \Re(z)\! \le\! 1$ and, in this range, let  $|\arg{F(z)}|<\pi$.
\vskip 4pt

These conditions allow to convert the contour integral into the ``solvable'' integral equation; of course, at the appropriate choice of the contour of integration.
Let contour $\Omega$ consists of the four segments:\\
{\bf A.} %integration 
Segment along line $\Re(t)\!=\!1$ from $t=1\!-\!\rmi A$ to $t=1\!+\!\rmi A$.\\
{\bf B.} Segment from point $t=1\!+\!\rmi A$ to $t=-\!1\!+\rmi A$, passing above point $z$.\\
{\bf C.} Segment along line $\Re(t)=-1$ from $t=-\!1\!+\rmi A$ to $t=-\!1\!-\!\rmi A$.\\
{\bf D.} Segment from point $t=-\!1\!-\!\rmi A$ to $t=1\!-\!\rmi A$, passing below point $z$.
\vskip 3pt

For this contour $\Omega$, the Cauchy integral can be written as follows:
\be 
\!\!\!\!\!\!\!\!\!\!
F(z)\!&=&\! \frac{1}{2\pi} \int_{-A}^{A} \frac{F(1\!+\!\mathrm i p) \, \mathrm{d} p}{1+\mathrm i p - z}
- \frac{1}{2\pi} \int_{-A}^{A} \frac{F(-1\!+\!\mathrm i p) \, \mathrm d p}{-1+\mathrm i p - z}
\nonumber \\ && \rule{322pt}{0pt} 
\label{3dot2} 
\\
&&- ~ \frac{F_{\rm up}}{2\pi\mathrm i} \int_{-1-\mathrm i A}^{1-\mathrm i A} \, \frac{ \mathrm d t}{t\!-\!z}
+ \frac{F_{\rm down}}{2\pi\mathrm i} \int_{-1-\mathrm i A}^{-1-\mathrm i A} \, \frac{ \mathrm d t}{t\!-\!z}
\nonumber \ee
where  $F_{\rm up}$ и $F_{\rm down}$ 
are some mean values of function $F$  in vicinity of the segments {\bf B} and {\bf D} of the contour $\Omega$.
\vskip 6pt

Taking into account the transfer equation \ref{transfereq}, 
and assuming holomorphism of function $T^{-1}$, equation  \rf{3dot2} can be rewritten as follows:
\be  \!\!\!\!\!\!\!\!\!\!
F(z)\!=\! \frac{1}{2\pi} \int_{-A}^{A} ~ \frac{\exp\!\big(F(\mathrm i p)\big) \, \mathrm{d} p}{1+\mathrm i p - z}
- \frac{1}{2\pi} \int_{-A}^{A} ~\frac{\ln\!\big(F(\mathrm i p)\big) \, \mathrm d p}{-1+\mathrm i p - z} + K(z)
~ ~ \label{3dot3}
\ee
where %где
\be  \!\!\!\!\!\!\!\!\!\!
K(z) &=& ~
F_{\rm up}  ~   \cdot \left( \frac{1}{2} - \frac{1}{2 \pi \mathrm i} \ln \frac{ 1-\mathrm i A +z}{1-\mathrm i A - z} \right)
\nonumber \\
&+&F_{\rm down} \cdot \left( \frac{1}{2} - \frac{1}{2 \pi \mathrm i} \ln \frac{ 1-\mathrm i A -z}{1-\mathrm i A + z} \right)
~ ~ \eL{K}
This representation implies that the modulus of phase of function $F$ along the imaginary axis remains less than $\pi$, so, 
the contour of integration does not cross the cut line of the logarithmic function in \rf{K}.
\vskip 6pt

Equations \rf{3dot3},\rf{K} are still exact.
However, they become approximations, if we replace
 $F_{\rm up}\!\rightarrow\! L$ and $F_{\rm down}\!\rightarrow\! L^*$.
This replacement leads to the closed representation for $K$.
We get the ``solvable'' integral equation for the approximation $F_A(\mathrm i y)$ of superfunction $F$
along the imaginary axis:
\be
F_A(\mathrm i y)= \frac{1}{2\pi} \int_{-A}^{A} ~ \frac{\exp\!\big(F_A(\mathrm i p)\big) \, \mathrm{d} p}{1+\mathrm i p - \mathrm i y} 
- \frac{1}{2\pi} \int_{-A}^{A} ~\frac{\ln\!\big(F_A(\mathrm i p)\big) \, \mathrm d p}{-1+\mathrm i p - \mathrm i y} 
+ K_A(\mathrm i y) 
\!\!\!\!\!\!\!\!
\nonumber \\
\label{3dot5}
\ee
where
\be 
K_A(z) =
L    \cdot \left( \frac{1}{2} - \frac{1}{2 \pi \mathrm i} \ln \frac{ 1\!-\!\mathrm i A \!+\!z}{1\!+\!\mathrm i A \!-\! z} \right)+
L^* \cdot \left( \frac{1}{2} - \frac{1}{2 \pi \mathrm i} \ln \frac{ 1\!-\!\mathrm i A \!-\!z}{1\!+\!\mathrm i A \!+\! z} \right)
\nonumber \\
\label{KA}
\ee
Equations  \rf{3dot5} и \rf{KA} include no unknown parameters; neither  $F_{\rm up}$ not $F_{\rm down}$ appear there.
In such a way, equations  \rf{3dot5} and \rf{KA} can be used to calculate $F_A$.
\vskip 6pt

Representation of the integral by the uppest and lowest pars of the contour $\Omega$ through function $K_A$ by  \rf{KA} is not trivial.
The first look at the expression causes the seduction to rewrite each logatightm as difference of two logarithms, in order to simplify the expression in the argument. However, in other to get a robust representation for tetration, this 
is not a good idea.
On the representation  \rf{KA}, the cuts are directed away from the imaginary axis; they go horizontally, parallel to the abscissa axis.
This helps to avoid approaching of the argument of the primary approximation of tetration to the 
boundaries of holomorphism of function 
$K_A$.
Complex map of function $K_A$ is shown in figure \ref{mapK} for $A\!=\!3$, $A\!=\!5$ and $A\!=\!10$ with lines
of constant logamplitude $\rho$ and constant phase $\phi$ such that $\exp(\rho\!+\!\mathrm i \varphi) \!=\! K_A(x\!+\mathrm i y)$ in the $x$, $y$ plane.

\begin{figure}[t]
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\put(77.7,17){\sx{.5}{$x$}} 
}

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\end{picture}}
\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Analuxp02t900.jpg}
\vskip -6pt
\caption{\large $\exp(\rho\!+\!\mathrm i \varphi) \!=\! K_A(x\!+\mathrm i y)$ 
by
\rf{KA} 
for
$A\!=\!3,5,10$ 
\label{mapK}}
\end{figure}
\vskip 6pt

Solution  $F_A$  of equation \rf{3dot5} can be approximated with the iterates, described below.
At  $A\gg1$, solution $F_A$ 
provides the good approximation for the superfunction $F\approx F_A$;
\be
F(z)=\lim_{A\rightarrow \infty} F_A(z)
\eL{tetF}

In order to get tetration
$\mathrm{tet}$, 
the additional condition
$\mathrm{tet}(0)\!=\!1$. 
should be satisfied. So, I define 
\be
\mathrm{tet}(z)=F(x_0+z)
\eL{tet}
where $x_0$ 
is solution of equation
$F(x_0)\!=\!1$. 
Value
$x_0$ 
depends on the initial condition at the iterations, and also on the order of update of the values of the discrete approximation of function $F_A$.
For the initial approximation by \rf{Fit6}, the resulting $x_0$ happens to be or order of $0.1\,$; and it is important, that it remains smaller than unity. 
\vskip 6pt

Formula \rf{tet} can be considered as ``constructive definition'' of tetration tet, with all reverences about the existence and the uniqueness of the limit, at $A\!\rightarrow\!\infty$ and the limit of infinite increase of the number of points for the discrete approximation of the integrals. From the point of view of the ``pure'' mathematics, such a ``definition'' deserves critics, but I expect, with time, the ``pure mathematicians'' present a more efficient and elegant way of constructive definition of tetration.
For a while, tetration may remain as solution $f$ of the transfer equation $\mathrm e^{f(z})\!=\!f(z\!+\!1)$ bounded in the strip $|\Re(z)|<1$ with additional condition $f(0)\!=\!1$.  
\vskip 6pt

For the approximation of limit in equation \rf{tetF}, some finite value of $A$ should be chosen.
Value of increment $k\!=\!L\!\approx 0.318\!+\!1.337\,\mathrm i\,$ in approximation \rf{Fit6} 
indicates, that for the complex double implementation, the reasonable value of constant $A$ should be of order of 20.
It happened, that for $A\!=\!24$, the residual is a little bit smaller, and this value is used for the primary numerical implementation
\footnote{Using the numerical implementation of the Cauchy integral for the first time, I did not guess the simple estimate through the increment $k$;
so I had to increase value of $A$ until the residual at the substitution of the primary approximation into the transfer equation \rf{transferexp} became of order of the rounding errors of the complex double arithmetics}.
\vskip 6pt

For the numerical implementation of equation  \rf{3dot5}, the integrals are replaced to their approximations with the Gauss-Legendre quadrature formula. Then, the resulting equation can be solved with iterations, updating values of the function one by one.
\vskip 6pt

The attempt of the parallel assignment of the new values (that is easy to program with the high-level programming languages) leaded to the diverging algorithm. In order to get the convergence, I update first the odd nodes, and then the even ones. Then, after some teens of iterates, the procedure provides the accurate solution with 14 significant figures; this precision is estimated, evaluating the residual in various tests of the internal self-consistency.
\footnote{following Axiom 4 (see Introduction), I made certain efforts trying to refute, negate the concepts of existence and uniqueness of tetration.}
\vskip 6pt

Solution of equation \rf{3dot5} approximates values of superfunction $F$ along the imaginary axis. Then, equation \rf{3dot3} 
extends this approximation to the strip
\be
-1<\Re(z)<1 
\eL{strip1}
Accuracy of this primary approximation is poor in vicinity of the edges of the strip. Therefore, for the numerical implementation of tetration,
the narrower strip is used,
\be
-\frac{1}{2}\le \Re(z) \le \frac{1}{2} 
\eL{striphalf}
applying formula
\be
F(z)=T^n\big(F(z\!-\!n)\big)
\eL{Tn}
for some appropriate integer $n$, positive or negative, dependently on the sign of 
$\Re(z)$. 
\vskip 6pt


At the increasing of value of parameter $A$ in equation \rf{3dot2}, function $F$ 
approaches to the solution of the transfer equation \rf{transferexp}. 
This solution does not yet satisfy the condition
$F(0)\!=\!1$, 
but, the required tetration appears as the appropriate displacement of the argument with equation \rf{tet}.
\vskip 6pt


In figure   \ref{mapK}, the strip $|x|\! \le \!0.5$, $|y|\! \le \!4.5~$ is shaded;
roughly, this is region, where the function $K_A$ is used at the evaluation of superfunction $F$ 
at the calculation of map in figure \ref{analuxp01}{\bf d}.
The complex maps % \ref{mapK} 
verify, that the phase of reconstructed function 
does not exceed 2; and therefore, does not exceed   $\pi$. 
This justifies use of logarithm in formula \rf{3dot3}.
However, this cannot be considered as a rigorous proof of existence and the uniqueness of the resulting function. I hope, the rigorous proof 
will be reported soon by the ``pure'' mathematicians. While I present only the computational evidence of the existence and the uniqueness.
\vskip 6pt

In general case, applying this method to general transfer function $T$ with complex fixed points, $F(z\!-\!n)$ could happen at the point of singularity of at the cut line of function $T$ or $T^{-1}$. 
This would indicate, that the resulting superfunction is also singular.
(For tetration, this happens at the real argument, equal to or smaller than $-2$.)
This may limit the range of applicability of the method suggested here.
\vskip 2mm

In the first calculus, the approximation $\mathrm{fit}_3$ had been used as the initial probe function for the iterational solution of equation \rf{3dot5}.
Then it happened, that the iterates with other (more primitive) probe functions lead to the same result, providing the same tetration.
With the algorithm above, in 2008, the natural tetration had been constricted \cite{analuxp}.
For $A\!=\!24$, the Gauss-Legendre quadrature formula with 2048 nodes gives the accurate approximation: at the substitution into equation \rf{transfereq}, it gives the residual of order of $10^{-14}$, while the variables complex(double) 
are used. Of order of 14 significant figures of the solution can be evaluated in real time.
This indicates stability of evaluation of tetration through the Cauchy integral.
\vskip 6pt

The algorithm above had been used to plot first maps and explicit plots of tetration to base 
 $b>\exp(1/\mathrm e)$
\cite{toritet,cztet}. In particular, this algorithm is used to plot the map at the bottom of figure \ref{analuxp01}. 
The direct representation through the Cauchy integral is fast enough to plot the maps and the explicit plots of tetration in real time.
However, it becomes slow, if the tetration is used for evaluation of other function, for example, its superfunction (pentation)
considered below in chapter 19. For the application of tetration, even faster approximations may have sense. One of them is considered in the next section.

\section{Taylor expansion at zero}
\baselineskip 19pt

I wanted the check the claim, that solution $f\!=\!\mathrm{tet}$ of equations \rf{transferexp} and \rf{transferexp},
that satisfies properties T1-T4, exists and is unique. As it is declared above, the numerical test does not substitute the rigorous proof, in the similar way, as the rigorous proof does not substitute the numerical tests. For the serious tests, it is important, that the function is fast to evaluate. In order to boost the evaluation, the approximations through the elementary functions had been suggested \cite{vladie}. One of them refers to the Taylor expansion at zero. It is considered in this section.
\vskip 6pt

Derivatives of tetration can be calculated by differentiation of the primary representation by  \rf{3dot5}.
More accurate values can be obtained with the Cauchy integral formula with the circular contour of integration
(assuming, that the primary representation is already implemented). Radius of this circle should be less than 2, and can be slightly larger than unity; then, the error of the result reduces due to denominator in the fraction under the integral in the right hand sidle of equation \rf{Cauchy}. In such a way, the coefficients  in the expansion below are evaluated.
\be \!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
\mathrm{naiv}(z)&=&\sum_{n=0}^{N-1}c_{n}z^{n} 
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
\iL{vladinaiv}
\\
\mathrm{tet}(z)&=&\mathrm{naiv}(z)+\mathcal{O}(z^{N})
~ ~ ~
\eL{vladinaiv2}
Evaluations of coefficients $c$ are shown in the first column of table
\ref{vladitab1}. 


\begin{table}[t]
\caption{ %Таблица \ref{vladitab1}. 
Coefficients in expansions \rf{vladinaiv}, \rf{maclo} and \rf{vladitai} \label{vladitab1}}

\hskip -6pt
\sx{.94}
{\begin{tabular}{rrrrr}
$n$& $c_{n}~~~~~~~~~$& $s_{n}~~~~~~~~~$& $\Re(t_{n})~~~~~~~~~$ & $\Im(t_n)_{\rule{0pt}{12pt}}~~~~~~~~~$\\ 
\input{vladitab1}
\end{tabular}}
\end{table}
\vskip 2mm

The Taylor expansion $\mathrm{tet}(z)$ at $z\!=\!0$ converges for 
$|z|\!<\!2$.
The radius of convergence is determined by the distance from the point of expansion (id est, from zero) to the nearest singularity, which is $-2$. 
\vskip 2mm

For the numerical implementation, the number of terms chosen 
$N\!=\!50$.
Complex map of the resulting naive approximation by 
\rf{vladinaiv}
is shown in the left hand side of figure 
\ref{vladi04map} 
with lines of constant real part and content imaginary part,
$u\!+\!\mathrm i v=\mathrm{naiv}(x\!+\!\mathrm i y)$. 
The thick lines show levels
$u\!=\!\Re(L)$ 
and
$v\!=\!\pm \Im(L)$.
\vskip 2mm

In order to 
verify the precision of the approximation $f\!=\!\mathrm{naiv}$,
the central and the right hand side pictures of figure \ref{vladi04} show the maps of agreements
 \be \!\!\!\!\!\!\!\!\!\!
D_{\rm naiv1}(z)=-\lg\left( \frac
{|\ln(\mathrm{naiv}(z\!+\!1)-\mathrm{naiv}(z)|}
{|\ln(\mathrm{naiv}(z\!+\!1)|+|\mathrm{naiv}(z)|} \right)
\label{vladienaivD1}
\ee
\be \!\!\!\!\!\!\!\!\!\!
D_{\rm naiv2}(z)=-\lg\left( \frac
{|\exp(\mathrm{naiv}(z\!-\!1)-\mathrm{naiv}(z)|}
{|\exp(\mathrm{naiv}(z\!-\!1)|+|\mathrm{naiv}(z)|} \right)
\label{vladienaivD2}
\ee
Functions of agreement $D$ indicate, how many significant figures of tetration can be expected to appear at evaluation of tetration with approximation ``naive'' by \rf{vladinaiv}. 
Levels 
$D\!=\!1,2,4,6,8,10,12,14$ are shown.
Level $D\!=\!1$ is shown with thick lines.
Symbol ``15'' indicates the domain, where the agreement is better than 14.
We may expect, at
$|z|\!<\!1$, 
the polynomial by 
\rf{vladinaiv} 
provides of order of 14 significant figures; this is close to the maximal precision for variables complex double.
\vskip 6pt

\newcommand \vladiax
{
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\put(-3,49){\sx{.5}{$2$}}
\put(-3,39){\sx{.5}{$1$}}
\put(-3,29){\sx{.5}{$0$}}
\put(-7,19){\sx{.5}{$-1$}}
\put(-7, 9){\sx{.5}{$-2$}}
\put( 6  ,-4){\sx{.5}{$-2$}}
\put(17  ,-4){\sx{.5}{$-1$}}
\put(30,-4){\sx{.5}{$0$}}
\put(40,  -4){\sx{.5}{$1$}}
\put(50,  -4){\sx{.5}{$2$}}
\put(58, -4){\sx{.5}{$x$}}
}
\begin{figure}[t]
\normalsize
\begin{center}
~\sx{2.02}{\begin{picture}(70,60)
\put(0,0){\includegraphics{vladi04a}}
\put(25,24){\sx{.4}{\rot{90} $ u\!=\!\Re(L)$ \ero }}
\put(32,51){\sx{.4}{\rot{-61} $ v\!=\!\Im(L)$ \ero }}
\put(35,11){\sx{.4}{\rot{61} $ v\!=\!\Im(L^*)$ \ero }}
\vladiax
\end{picture}}
\sx{2.02}{\begin{picture}(70,60)
\put(0,0){\includegraphics{vladi04b}}
\vladiax
\put(23,29){\sx{.55}{$15$}}
\put(43, 53){\sx{.55}{$D_{1}\!<\!1$}}
\end{picture}}
\sx{2.02}{\begin{picture}(58,60)
\put(0,0){\includegraphics{vladi04c}}
\vladiax
\put(32,29){\sx{.55}{$15$}}
\put(43,53){\sx{.55}{$D_{2}\!<\!1$}}
\end{picture}}
\end{center}
\vskip -4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi04.jpg}
\vskip -2pt
\caption{ %\large
$u\!+\!\mathrm i v\!=\!\mathrm{naiv}(x\!+\!\mathrm i y)$ 
by \rf{vladinaiv} for $N\!=\!50$, 
left; agreements
$D_{1}\!=\!D_{\rm naiv1}(x\!+\mathrm i y)$ 
and
$D_{2}\!=\!D_{\rm naiv2}(x\!+\mathrm i y)$ 
by
\rf{vladienaivD1} 
and
\rf{vladienaivD2},
central and right hand side maps.
\iL{vladi04} 
\label{vladi04map}
}
\vskip 14pt
\end{figure}

Evaluations with 50 terms is considered for verification of the expansion.
At the evaluation of tetration, for example, at the implementation for real argument, the number of terms can be significantly reduced without loss of precision.
\vskip 8pt

The polynomial approximation $\mathrm{naiv}$ by \rf{vladinaiv}
can be used for the precise and fast evaluation of tetration, while the modulus of its argument is smaller or of order of unity.
For the efficient implementation, this is good, but it is not sufficient. In the next section, the advanced expansion is considered, that allows to extend the range of the approximation of tetration for moderate values of the argument.

\section{Improved approximation}
\baselineskip 19pt

The range of the accurate approximation of tetration can be extended, if we take into account the logarithmic singularity of tetration.
I  ``switch-out'' the singularity at $-2$, expanding function $\tet(z)-\log(z\!+\!2)$ instead of $\tet(z)$. 
This expansion gives the approximation below, I call it ``maclo'':
\be
\mathrm{maclo}(z)&=&\ln(z\!+\!2)
+\sum_{n=0}^{N-1} s_{n} z^n~;~~~
\iL{maclo} \\
\tet(z)&=&\mathrm{maclo}(z)+\mathcal{O}(z^{N})
~.~\eL{macloN}
For $n=101$, function maclo is shown in the left map of figure  \ref{vladi05}.
\vskip 2mm

The series, used for approximation \rf{maclo} converges at $|z|\!<\!3$;
the function reproduces the logarithmic branch point and even part of the cut at $z\!<\!-2$.
Approximate values of first coefficients $s$ are shown in the second column of table \ref{vladitab1}. 
\vskip 2mm

\newcommand \vladimaclo {
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\put(-3,49){\sx{.5}{$2$}}
\put(-3,39){\sx{.5}{$1$}}
\put(-3,29){\sx{.5}{$0$}}
\put(-5,19){\sx{.5}{$-\!1$}}
\put(-5, 9){\sx{.5}{$-\!2$}}
\put(55 , -4){\sx{.45}{$\Re(z)$}}
\put(50 , -4){\sx{.5}{$2$}}
\put(29.5,-4){\sx{.5}{$0$}}
\put( 6  ,-4){\sx{.5}{$-\!2$}}
}
\begin{figure}[t]
\begin{center}
~
\sx{2}{\begin{picture}(68,60)
\normalsize
\put(0,0){\includegraphics{vladi05a}}

\put(25,24){\sx{.4}{\rot{90} $ u\!=\!\Re(L)$ \ero }}
\put(32.6,27){\sx{.4}{\rot{90} $ u\!=\!1$ \ero }}
\put(31,56){\sx{.4}{\rot{-67} $ v\!=\!\Im(L)$ \ero }}
\put(33,5){\sx{.4}{\rot{67} $ v\!=\!\Im(L^*)$ \ero }}
\put(25,44.6){\sx{.4}{\rot{-31} $ v\!=\!1$ \ero }}
\put(28,16){\sx{.4}{\rot{37} $ v\!=\!-1$ \ero }}

\vladiax
\end{picture}} ~
\sx{2}{\begin{picture}(68,60)
\put(0,0){\includegraphics{vladi05b}}
\vladiax
\put(14,33){\sx{.52}{$D_{3}>14$}}
\put(43, 4){\sx{.52}{$D_{3}\!<\!1$}}
\end{picture}} ~
\sx{2}{\begin{picture}(60,60)
\put(0,0){\includegraphics{vladi05c}}
\vladiax
\put(23,33){\sx{.52}{$D_{4}>14$}}
\put(44, 3){\sx{.52}{$D_{4}\!<\!1$}}
\end{picture}}
\end{center}
\vskip -8pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi05.jpg}
\vskip -4pt
\caption{
\large 
$\mathrm u\!+\!\mathrm i v \!= \! \mathrm{maclo}(x\!+\! \mathrm i y)$  
by \rf{maclo} 
at
$N\!=\!101$, 
left;
agreements
$D_3$ и $D_4$ 
by
\rf{macloE} и \rf{macloL}, 
centre and right
\label{vladi05}
\label{figmaclo}
}
\vskip 14pt
\end{figure}

The range of approximation of tetration $\tet$ with function $\mathrm{maclo}$ is 
wider, than that by the Taylor expansion of tetration at zero;
compare figure \ref{vladi04} and figure \ref{vladi05}.
The central and right hand side maps of figure show agreements
\be \!\!\!\!\!\!
D_{3}(z)&=&-\lg\! \left( \frac
{\Big| \ln\!\big(\mathrm{maclo}(z\!+\!1)\big)-\mathrm{maclo}(z)\Big|}
{\Big| \ln\!\big(\mathrm{maclo}(z\!+\!1)\big)\Big|+\Big|\mathrm{maclo}(z)\Big |} \right)
~~ \label{macloE}\\
\!\!\!\!\!\!
D_{4}(z)&=&-\lg\! \left( \frac
{\Big| \exp\!\big(\mathrm{maclo}(z\!-\!1)\big)-\mathrm{maclo}(z)\Big|}
{\Big| \exp\!\big(\mathrm{maclo}(z\!-\!1)\big)\Big|+\Big|\mathrm{maclo}(z)\Big |} \right)
~~
\label{macloL}
\ee
Within the central loop, the residuals at the substitution
$f\!\rightarrow\!\mathrm{maclo}$
into equations
\rf{transferexp}
are of order of
$10^{-15}$.
\vskip 6pt

While $|z|\!<\!2$, the approximation $\mathrm{maclo}(z)$ with a hundred terms provides of order to 14 significant figures of tetration $\mathrm{tet}(z)$.
However, while the module of the argument increases and becomes larger than two, the accuracy of this approximation quickly drops down. 
In order to even extend the range of the fast approximation, the Taylor expansion at some point at the imaginary axis can be used. In the next section, 
the Taylor expansion at point 
$3\,\mathrm i$ is described.

\section{Expansion of $\mathrm{tet}(z)$ at  $z\!=\!3\,\mathrm i$}
\baselineskip 20pt

For evaluation of tetration, we should cover some strip of unity width along the imaginary axis with 
good (fast and precise) approximations. The approximation maclo from the previous section
does not approximate tetration at point  $3\,\mathrm i$. For me, this is sufficient reason (or, may be, a pretext)
to prepare the Taylor expansion of tetration namely in this point
\footnote{Constant 3 appears as minimal integer number for which (with coefficient i) approximation maclo by \rf{maclo} fails.}.
This expansion is described below.
\vskip 2mm

The truncated Taylor expansion of  $\mathrm{tet}(z)$ at point  $z\!=\!3\,\rmi$  is denoted with name ``tai''
(TAylor expansion centered at the Imaginary axis):
\be
\mathrm{tai}(z)=\sum_{n=0}^{N-1} t_{n}\left(z\!-\!3\,\mathrm i\right)^n
\label{tai}
\eL{vladitai} 
Approximations of the coefficients $t$ are calculated with the Cauchy integral. The real and imaginary parts of the first coefficients are presented in the last two columns of table
\ref{vladitab1}. 
The series converges at
$|z\!-\!3\mathrm i |\!<\! \sqrt{2^2\!+\!3^2}\!=\!\sqrt{13}\!
\approx\! 3.6~
$. 
For the numerical implementation I choose value $N\!=\!51$; then, at $|z\!-\!3\,\mathrm i |<2$,
approximation tai by \rf{vladitai} provides of order of 14 significant figures.
The complex map or this approximation is shown in the left hand side of figure 
\ref{vladi06}. 


\newcommand \taiax {
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\put(-3,69){\sx{.5}{$6$}}
\put(-3,59){\sx{.5}{$5$}}
\put(-3,49){\sx{.5}{$4$}}
\put(-3,39){\sx{.5}{$3$}}
\put(-3,29){\sx{.5}{$2$}}
\put(-3,19){\sx{.5}{$1$}}
\put(-3, 9){\sx{.5}{$0$}}
\put(-5,-1){\sx{.5}{$-\!1$}}
\put( 7  ,-4){\sx{.5}{$-\!3$}}
\put(17  ,-4){\sx{.5}{$-\!2$}}
\put(27  ,-4){\sx{.5}{$-\!1$}}
\put(40 , -4){\sx{.5}{$0$}}
\put(50 , -4){\sx{.5}{$1$}}
\put(60 , -4){\sx{.5}{$2$}}
\put(70 , -4){\sx{.5}{$3$}}
\put(79 , -4){\sx{.5}{$x$}}
}
\begin{figure}[t]
\vskip 22pt
\hskip 12pt
\sx{2.23}{\begin{picture}(100,80)
\put(0,0){\includegraphics{vladi06a}}

\put(13,55){\sx{.44}{\rot{-77} $ v\!=\!\Im(L)$ \ero }}
\put(23,62.4){\sx{.44}{\rot{-77} $ u\!=\!\Re(L)$ \ero }}
\put(34,65){\sx{.44}{\rot{-76} $ v\!=\!\Im(L)$ \ero }}
\put(45.6,66){\sx{.44}{\rot{-76} $ u\!=\!\Re(L)$ \ero }}
\put(57.6,65){\sx{.44}{\rot{-77} $ v\!=\!\Im(L)$ \ero }}

\put(42,7){\sx{.4}{\rot{86} $ u\!=\!1$ \ero }}
\put(35,24.6){\sx{.4}{\rot{-31} $ v\!=\!1$ \ero }}

\taiax
\end{picture}}
\sx{2.23}{\begin{picture}(80,80)
\put(0,0){\includegraphics{vladi06b}}
\taiax
\put(26,40){\sx{.58}{$D_{5}\!>\!14$}}
\put(62,15){\sx{.58}{$D_{5}\!<\!1$}}
\end{picture}}
\vskip 8pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi06.jpg}
\vskip -2pt
\caption{ $u\!+\!\mathrm i v\!=\!\mathrm{tai}(x\!+\!\mathrm i y)$ 
by
\rf{tai}
{at}
$N\!=\!51$
and agreement
$D_{5}(x\!+\!\mathrm i y)$
\label{vladi06}
\label{figtai} }
\vskip 10pt
\end{figure}

The right hand side map in figure \ref{vladi06} shows the agreement
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
D_5(z) &=&-\lg\! \left( \frac
{\Big| \ln\!\big(\mathrm{tai}(z\!+\!1)\big)-\mathrm{tai}(z)\Big|}
{\Big| \ln\!\big(\mathrm{tai}(z\!+\!1)\big)\Big|+\Big|\mathrm{tai}(z)\Big |} \right)
~ 
\eL{vladiD5}
As in figures  \ref{vladi04}, \ref{vladi05}, and \ref{vladi06}, the levels for the agreement are drown with increment 2, beginning with 2; one additional level $D_5\!=\!1$ is shown with thick line. Inside the inner loop, the agreement with at least 14 digits takes place.
\vskip 2mm

Approximation tai by  \rf{vladitai} significantly extends the domain, where the tetration can be precisely evaluated through elementary functions. For positive values of  $\Im(z)$, tetration $\mathrm{tet}(z)$ can be approximated with
\be
\mathrm{tet}(z)\approx \mathrm{tai}(z)
\ee
For negative $\Im(z)$, tertian can be approximated with
\be
\mathrm{tet}(z)\approx \mathrm{tai}(z^*)^*
\ee
These representations are sufficient to plot map in figure  \ref{analuxp01}d.
I assume, that the transfer equation \rf{transferexp} is applied some integer number of times, in order 
use $\mathrm{tai}(z)$ with the argument from the strip $|\Re(z)| \!\le \!1/2$. 
However, the expansions above  do not provide the accurate approximation of $\mathrm{tet}(z)$ at 
$|\Im(z)|\!>\!5$.
\vskip 2mm

One could extend the range of approximation, using the truncated Taylor expansions at point $5\, \mathrm i$ (or even $6 \,\mathrm i$), this would significantly extend the range of approximation, and continue such an exercise with new and new points along the imaginary axis.
However, there exist more intelligent and elegant way to deal with cases, when the imaginary part of the argument is large.
This way is described in the next section.

\section{Asymptotic expansion
}
\baselineskip 18pt
The approximation of $\mathrm{tet}$ at large values of its argument can be build up using the asymptotic representation
\be
\tet_{\cal A}(z)=L+\sum_{n,m} \mathcal{A}_{m,n} \exp\left( L n z + \alpha m z \right)
\eL{fimao}
$L\!\approx\! 0.31813150520476413 \!+\! 1.3372357014306895 \,\mathrm i$ 
is, as before, the fixed point of logarithm, $L\!=\!\ln(L)$, 
and $\mathcal A$ are constant coefficients.
\vskip 6pt

Substitution $f=\tet_{\cal A}$ into the transfer equation  \rf{transferexp} 
gives the chain of equations for coefficients  $\mathcal A$.
These equations do not determine ${\cal A}_{m,0}$; so, the solution still has the countable set of ``free'' parameters for natural $m$.
Difficulty of determination of these parameters had been discussed in 1950 by Hellmuth Kneser \cite{kneser}.
However, even a relatively small amount of terms taken into account in expansion \rf{fimao}
can be used for the precise approximation and evaluation of tetration at large values of the imaginary part of the argument.
\vskip 6pt

Looking at the general (and a little bit ugly) expansion  \rf{fimao}, I suggest the approximate, but more beautiful formula
\newcommand \ve {\varepsilon}
\be 
{\rm fima}(z)=\sum_{n=0}^N a_n \ve^n +
\beta \ve \exp(2 \pi \rmi z)%\sum_{m=0}^M b_m \ve^m
~,~\eL{fima}
where the small parameter
\be
\ve\!\!=\!\exp(Lz\!+\!Lr)
\eL{fimave}
Mnemonics of name fima is following: Functional expansion for large IMAginary part of the argument.
Substitution of
$f(z)\!=\!\mathrm{fima}(z)\!+\!O(\ve^{N+1})$
into the transfer equation  \rf{transferexp} gives the coefficients
\be \!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!a_0\!&\!=\!&~L~ ~\approx~ 
   0.31813150520%47641353 
+ 1.33723570143%06894089 
\, \rmi\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!a_1\!&\!=\!&~1~\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!a_2\!&\!=\!&\!\frac{1/2}{L-1}~\approx~ 
-0.1513148971%5565%17359
-0.2967488367%3224%13067
\, \rmi\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!a_3\!&\!=\!&\!\frac{a_{2}+1/6}{L^2-1}=\frac{2+L}{6(L\!-\!1)(L^2\!-\!1)} \approx \! 	
      - 0.036976%3094%0906762
\! +\! 0.098730%5443%1149697 
\, \rmi 
~~~~~~~\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!a_4\!&\!=\!&\!\!\frac{6+6L+5L^2+L^3}{24(L\!-\!1)^3(L\!+\!1)(L^2\!+\!L\!+\!1)}\approx 
      0.02581%15979731%401398
 \!-\!0.01738%69621265%30755
\, \rmi~~ ~ ~ ~ ~\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!a_5\!&\!=\!&\!\frac{24\!+\!36L\!+\!46 L^2\!+\!40L^3 \!+\! 24 L^4 \!+\! 9 L^5 \!+\! L^6} 
                   {120(L\!-\!1)^4(L\!+\!1)^2(1\!+\!L+\!2L^2\!+\!L^3\!+\!L^4)}   \nonumber \\ 
                  & \approx&  
      -0.0079444196 
\!+\! 0.00057925018 
\, \rmi 
\label{fimavalua}
\ee
It is not difficult to take into account more terms, but $N\!=\!5$ already allows to cover the rest of the complex plane (not covered with approximations ``maclo'' and ``tai'') with accurate approximations of natural tetration.
\vskip 8pt

Coefficients  $R$ and $\beta$ in the right hand sides of formulas \rf{fima} and \rf{fimave} remain as ``adjusting parameters''.
Their values are chosen in order to approximate tetration, evaluated with a little bit slower Cauchy integral:
\be 
\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!
r&\approx&  1.0779614375280 - 0.94654096394782\,\mathrm i  ~ ~ ~ \iL{fimaR}~~~~\\
\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!
\beta&\approx& 0.12233176 - 0.02366108 ~ ~ ~ \mathrm i 
~~~~~~~~~ ~~~~~~~
\iL{fimaB}~~~~
\ee
These values can be interpreted as approximations of the fundamental mathematical constants.
I suggest to call them ``the Kneser constants'', as the expansion with these coefficients had been suggested in 1950 by H.Kneser \cite{kneser}. Many digits in approximations of these constants can be calculated, in a way, similar to that in centuries 19 and 20 the mathematicians competed in precision of evaluation of number $\pi$.
\vskip 6pt

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\begin{figure}[t]
\hskip -40pt
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\put(-1,-7){\includegraphics{vladi03a}}
\multiput(27,38)(22.7,4){4}{\sx{.4}{\rot{-76} $v\!=\!\Im(L)$\ero}}
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\fimax
\end{picture}}

\hskip -40pt
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\put(-1,-7){\includegraphics{vladi03b}}
\fimax
\put( 64,48){\sx{.6}{$D_{\rm fifi}>14$}}
\put(151,8){\sx{.6}{$D_{\rm fifi}\!<\!1$}}
\end{picture}}
\vskip -2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi03.jpg}
\vskip -4pt
\caption{%\large
Top: $u\!+\!\mathrm i v\!=\!\mathrm{fima}(x\!+\!\mathrm i y)$ 
by
\rf{fima};
bottom: map of
$D_{\rm fifi}=D_{\rm fifi}(x\!+\!\mathrm i y)$  
by
\rf{fifi} %\rf{fimaE} 
\label{vladi03}
\iL{figfima} }
\vskip 4pt
\end{figure}
Complex map of function fima is shown in the top picture of figure  \ref{vladi03}; the upper half of the complex plane is shown. This map should be compared to the map of tetration in figure \ref{analuxpmap} (although the levels 
$\Re(L)$ 
and $\Im(L)$ are not drawn in figure \ref{analuxpmap}) and to maps of other approximations in figures 
\ref{vladi04},
\ref{vladi05} and
\ref{vladi06}
\vskip 8pt

There is no fundamental limit on the precision of evaluation of tetration (for example, through the Cauchy integral), so, 
parameters $\beta$ and $r$ in should be considered as fundamental mathematical constants.
The numerical computations, described in this book, have precision of order of 14 decimal digits (that is close to the best precision achievable with variable complex double), and parameter $r$ is evaluated with the similar precision. Precision of evaluation of parameter $\beta$ is not so high; perhaps, calculus with variables ``long complex double'' are necessary to improve the precision of evaluation of $\beta$ and add more digits in the right hand side of equation \rf{fimaB}. 
\vskip 2mm

In order to show the residual at the substitution $f\!=\!\mathrm{fima}$ into the transfer equation  \rf{transferexp}, figure
\rf{vladi03} shows the agreement
\be
\!\!\!\!\!\!\!\!\!\!
D_{\mathrm {fifi}}(z)=-\lg \left( \frac
{ | \mathrm{fima}(z)-\exp(\mathrm{fima}(z\!-\!1)) | }
{ | \mathrm{fima}(z)|+|\exp(\mathrm{fima}(z\!-\!1)) | }
\right)
\eL{fifi}
This agreement can be considered as an estimate, for how many orders of magnitude the value of the function is larger, than the error of its evaluation with approximation fima. 
As in the previous maps of agreement, the levels are drown with interval two orders of magnitude; only for level $D_{\mathrm{fifi}}\!=\!1$, the exception is done; this level is shown with thick line. Below this level, the approximation fima does not reproduce even the qualitative behaviour of natural tetration. In the upper region, above the highest level, contrary, the approximation provides at least 14 significant figures, that is close to the maximal precision, achievable with variables complex double.
\vskip 2mm

This section suggests the asymptotic approximation denoted with name ``fima'' by \rf{fima}. Approximation fima is valid in the most of the upper part of the complex plane. Its conjugation
$z\mapsto \mathrm{fima}(z^*)^*$  provides the approximation for the most of the lower part of the complex plane.
With the transfer equation, these approximations can be extended also to the larger values of the real part of the argument.
Together with approximations ``maclo'' and ``tai'', the whole complex plane happens to be covered with overlapping regions, and for each of these region, the efficient approximation based on the series expansion, is described. Now it would be methodically correct to analyse, verify the overlappings, agreement of these approximations. This overlapping is considered in the next section. 
\vskip 2mm

\section{Comparison of approximations}
\baselineskip 18.6pt
On the base of representation of natural tetration through the Cauchy integral, 
the coefficients of various expansions of tetration are evaluated and the approximations with elementary functions are suggested.
In this section, the mutual agreement or these representations is analysed.
\vskip 2mm


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\begin{figure}
\begin{center}
\sx{2.33}{\begin{picture}(96,75)
\put(0,0){\includegraphics{vladi07a}}
\tafiax
\put(31,54){\sx{.45}{$D\!>\!14$}}
\end{picture}}
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\put(0,0){\includegraphics{vladi07b}}
\tafiax
\put(33,55){\sx{.52}{$D\!<\!1$}}
\put(33,24){\sx{.45}{$D\!>\!14$}}
\end{picture}}
\end{center}
\vskip -18pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi07.jpg}
\vskip -4pt
\caption{
Comparison of approximations tai by (\ref{tai}) to fima by 
(\ref{fima})
and to maclo by (\ref{maclo}): agreements $D\!=\!D_{6}(x\!+\!\mathrm i y)$ and $D\!=\!D_{7}(x\!+\!\mathrm i y)$ by 
(\ref{D6}),(\ref{D7})
 in the complex z-plane.
\iL{figco}
~ ~ ~ ~ ~ ~
\iL{vladi07}
}
\vskip 8pt
\end{figure}

The left hand side of figure \ref{vladi07} 
shows the agreement of approximation tai by \rf{tai} with approximation fima by \rf{fima}:
\be
D_6(z)= - \ln \left( \frac
{|\mathrm{tai}(z) - \mathrm{fima}(z)|}
{|\mathrm{tai}(z)|+|\mathrm{fima}(z)|}
\right)
\label{D6}
\eL{vladiD6}
The right hand side of figure \ref{vladi07} shows agreement of approximation tai by \rf{tai} 
with approximation maclo  by \rf{maclo}"
\be
D_7(z)= - \ln \left( \frac
{|\mathrm{tai}(z) - \mathrm{maclo}(z)|}
{|\mathrm{tai}(z)|+|\mathrm{maclo}(z)|}
\right)
\label{D7}
\eL{vladiD7}

Figure \ref{figco}
indicates, how to choose the appropriate approximation of tetration dependently on the imaginary part of the argument $z$
at moderated values of $|\Re(z)|\!<\!1$. 
The boundary between the domains of the approximations should go through the loops, where 
$D>14$.
While $|\Im(z)|\le 1.5$, 
let approximation maclo be used;
At $1.5< \Im(z) \le 4.5$, 
let the approximation tai be used, and, at even larger values, let the evaluation be performed with approximation fima.

\EN{Looking at figure \ref{figco}, I suggest the following approximation:}% 
\be \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
{\rm fse}(z)=\left\{ \begin{array}{ccrcccr} 
{\rm fima}(z)		&,&	 4.5&	\!<\!	&\Im(z)			\\
{\rm tai}(z)		&,&	 1.5&	\!<\!	&\Im(z)	&\!\le\!&	 4.5	\\
{\rm maclo}(z)		&,&	-1.5&	\le	&\Im(z) &\!\le\!&	 1.5	\\ 
{\rm tai}(z^{*})^{*}	&,&	-4.5&	\le	&\Im(z) &\!<\!&\!\!	-1.5	\\ 
{\rm fima}(z^{*})^{*}	&,&	&		&\Im(z)	&\!<\!&\!\!	-4.5  
\end{array}
\right.
\label{fse}
\eL{fsexp}
This approximation can be compared to previous results.
The left hand picture of figure
\ref{figcoF}
shows  the agreement
\be
D_{8}(z)=-\lg\left( \frac
{|\mathrm{fse}(z)-F_{4}(z)|}
{|\mathrm{fse}(z)|+|F_{4}(z)|}
\right)
\eL{DfseF4}
of approximation $\mathrm{fse}$ with the approximation $F_{4}$ obtained through the direct implementation of the contour integral.%\cite{k}.
\vskip 2mm

\begin{figure}
\begin{center}
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\put(0,0){\includegraphics{vladi08a}}
\put(5,68){\sx{.45}{$\Im(z)$}}
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\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
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\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(72 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17  ,6){\sx{.5}{$-\!2$}}
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\end{center}
\vskip -16pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi08.jpg}
\vskip -4pt
\caption{Agreement $D\!=\!D_{8}$ by (\ref{DfseF4}), left;
the similar agreement for the contour integral with base domain shifted for $-0.5$. \iL{vladi08}\label{figcoF}}
\vskip 6mm
\end{figure}
Figure \ref{figcoF}
reveals the defects of each approximation. The jumps at
$\Im(z)\!=\!1.5$ and at $\Im(z)\!=\!2.5$
should be attributed to the transition from function
$\mathrm{maclo}$ 
to function $\mathrm{tai}$ 
and from function $\mathrm{tai}$
to function $\mathrm{fima}$ 
in the combination $\mathrm{FSE}$. 
Jumps at half-integer values of $\Re(z)$ should be attributed to the discontinuities of function $F_{4}$,
which extends the approximation with the contour integral, valid for $|\Re(z)|<1$, from the interval $|\Re(z)|\le 1/2$.
The rounding errors appear as irregular dots.
Within the strip $|\Re(z)|<1.5$,
the irregularities of all three approximations are of order of $10^{-14}$.
\vskip 2mm

The goal is to cover with efficient (fast and accurate) approximations at least the strip
$\Re(z)\le 0.5$;
then, values of natural tetration for the whole complex plane can be expressed through the transfer equation  \rf{transferexp} in the right hand side of the complex plane, and through the `inverted'' equation
\be
\ln( \mathrm{tet}(z))=\mathrm{tet}(z\!-\!1)
\eL{rtanexp}
in the right hand side.
The left map in figure \rf{vladi08} indicates, that the goal is achieved; agreement with approximately 14 decimal digits takes place in significantly wider part of the complex plane.
The approximations above are used for the fast implementation. 
\vskip 2mm

After to see the agreement discussed above, I had declared, that since now, the natural tetration can be evaluated so fast and so precisely, as other special functions, known since century 20.
Then Henryk Trappmann asked me to make one additional numerical test. He wanted to see, if the same tetration can be evaluated, if I misplpace the contour $\Omega$ in the Cauchy integral \cite{Cauchy}, moving it to the right. I recognised this as a trap (which would correspond to the last name of Henryk): if we displace the contour to the right, the derivatives of tetration becomes larger, and, with the same algorithm, we get lower precision.  But I agreed to displace the contour for 1/2 to the left. 
\vskip 2mm

With the displaced contour, the same ab initio evaluation of tetration had been performed. 
Tetration $\mathrm{tet}(-1/2\!+\!iy)$ for real $y$ had been evaluated; then, with the Cauchy integral and equations 
\rf{transferexp},\rf{rtanexp}, the approximation had been extended to the whole complex plane, in the similar way as with the first algorithm of evaluation of tetration \cite{analuxp}. The result is compared to the approximations with expansions in the way, similar to that of by  \rf{DfseF4}; the new approximation is used instead of $F_4$. The resulting agreement is shown in the right hand side map in figure \rf{vladi08}.
\vskip 2mm

Figure \ref{vladi08} reveals defects of approximations mentioned above. The discontinuities in formula \rf{fsexp} are seen with horizontal jumps along lines 
$\Im(z)=1.5$ and $\Im(z)=4.5$.% , that are clearly shown with concentrated levels.
Discontinuities of the initial, ``primary'' approximation appear with the vertical jumps along half-integer values of $\Re(z)$.
The similar discontinuities are seen also for the evaluation with displaced contour at integer values of $\Re(z)$. All these jumps of the compared approximations are at the level of $10^{-14}$, and this confirms the declared estimate of the precision of the evaluation of the natural tetration.
\vskip 2mm

The agreement in the right hand side of figure  \ref{vladi08} happened to be even better, than that in the left hand side. Henryk had been satisfied with that test. For the natural tetration, the new contour of integration happened to be a little bit better, than the initial choice.
In such a way, the analogy with lazy engineer (or with stupid selfish tyrant), mentioned above, gets the confirmation: the initial contour of integration in the original publication \cite{analuxp} is not best. However, I still think, that the simplicity of that contour and the good agreement (figure \ref{vladi08}) should be considered as some kind of excuse for the voluntaristic choice of the contour.
\vskip 2mm
 
\section{Implementation}

After the tests, described in the previous section, for the numerical implementation, the following approximation is used: 
$\mathrm{tet}(z)\approx \mathrm{FSE}(z)$, with
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{\rm FSE}(z)=\left\{ \begin{array}{ccrcccr} 
{\rm FIMA}(z)		&,&	 4.5&	<	&\Im(z)			\\
{\rm TAI}(z)		&,&	 1.5&	<	&\Im(z)	&\le&	 4.5	\\
{\rm MACLO}(z)		&,&	-1.5&	\le	&\Im(z) &\le&	 1.5	\\ 
{\rm TAI}(z^{*})^{*}	&,&	-4.5&	\le	&\Im(z) &<&\!\!	-1.5	\\ 
{\rm FIMA}(z^{*})^{*}	&,&	&		&\Im(z)	&<&\!\!	-4.5  
\end{array}
\right.
\label{FSEXP}
\eL{FSE}
\EN{where}\RU{где} 
\be \!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{\rm FIMA}&\!=\!&
\left\{\!
\begin{array}{ccc}
		\mathrm{fima}(z)		&,&\!	\Im(z) 	>	4\!+\!0.2379\!~ \Re(z)\\
		\exp( {\rm FIMA}(z\!-\!1) )\!\!\!\!	&,&\!	\Im(z)	\le	4\!+\!0.2379\!~ \Re(z)
\end{array}
\right.
\label{FIMA}
\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{\rm TAI}&=&
\left\{
\begin{array}{ccccr}
		\mathrm{tai}(z)				&,&\!	|\Re(z)|\!	&\le&	0.5 \\
		~\log({\rm TAI}(z\!+\!1))	&,&	\Re(z)		&<&	-0.5 \\
		~\exp({\rm TAI}(z\!-\!1))	&,&	\Re(z)		&>&	0.5 
\end{array}
\right. \iL{TAI}
\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{\rm MACLO}&=&
\left\{
\begin{array}{ccccr}
		\mathrm{tai}(z)					&,&\!	|\Re(z)|\!	\!&\!\le\!&\!	0.5 \\
		\log({\rm MACLO}(z\!+\!1))\!\!\!\!	&,&	\Re(z)		\!&\!<\!&\!	-0.5 \\
		\exp({\rm MACLO}(z\!-\!1))\!\!\!\!	&,&	\Re(z)		\!&\!>\!&\!	0.5 
\end{array}
\right. \label{MACLO}
\ee

This approximation provides of order of 14 correct significant figures of the holomorphic tetration $\mathrm{tet}$
and agrees with the previous results
\cite{analuxp}.
Up to my knowledge, up to year 2016, function $\mathrm{FSE}$ above is the most precise and the fastest among ever reported approximations of the tetrational.
However, in 2016-2017, the new results by William Paulsen, Samuel Cowgill appear;
they are not included here. In such a way, the numerical implementation of tetration can be improved. 

Mnemonics of the name FSE is obvious: Fast Super Exponent. 
The C++ implementation of this algorithm is loaded as \url{https://mizugadro.mydns.jp/t/index.php/Fsexp.cin} ; 
this approximation is used to plot the detailed map of tetration in figure 
\ref{tetmap}, used at cover of this Book.
\vskip 2mm

\begin{figure}[t]
\sx{.5}{\begin{picture}(852,820)
\put(40,20){\ing{b271tMap3}}
\zoomax
\put(60,415){\sx{2.8}{\bf cut}}
\put(340,548){\sx{3}{$v\!=\!1$}}
\put(360,414){\sx{3}{$v\!=\!0$}}
\put(760,414){\sx{3}{$v\!=\!0$}}
\put(340,280){\sx{3}{$v\!=\!-1$}}
\put(112,636){\sx{2.5}{\rot{-21}$v\!=\!1.4$\ero}}
\put(560,742){\sx{2.5}{\rot{-21}$v\!=\!1.4$\ero}}
\put(112,198){\sx{2.5}{\rot{20}$v\!=\!-\!1.4$\ero}}
\put(552, 90){\sx{2.5}{\rot{20}$v\!=\!-\!1.4$\ero}}
\put(662,716){\sx{2.5}{$u\!=\!0.2$}}
\put(216,606){\sx{2.5}{$u\!=\!0.2$}}
\put(660,642){\sx{3}{$u\!=\!0$}}
\put(210,536){\sx{3}{$u\!=\!0$}}
\put(386,628){\sx{2.5}{\rot{48}$u\!=\!0.4$\ero}}
\put(247,216){\sx{2.5}{\rot{23}$u\!=\!0.2$\ero}}
\put(480,140){\sx{2.5}{\rot{24}$u\!=\!0.4$\ero}}
\multiput(80,54)(232,-20){2}{\sx{3}{\rot{75}{$u\!=\!\Re(L)$}\ero}}
\multiput(190,30)(232,-20){2}{\sx{3}{\rot{75}{$v\!=\!\Im(L^*)$}\ero}}
\end{picture}}
\vskip 2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Tetma.jpg}
\vskip -4pt
\caption{ $u\!+\!\mathrm i v\!=\! \mathrm{tet}(x\!+\!\mathrm i y)$ 
\iL{tetmap}}
\end{figure}

Many terms are kept in approximations (\ref{tai})
and (\ref{maclo}) 
in order to provide the wide range of the overlapping in figures
\ref{figco} and \ref{figcoF}.
At the final step of the implementation, the number of terms can be reduced, boosting the algorithm, without loss of the precision.
In particular, this applies to the evaluation of tetration along the real axis: it is sufficient to approximate
$\mathrm{tet}(z)$
for
$|z|\le 1/2$,
using only a quarter of the radius of the precise approximation with function  $\mathrm{maclo}$. 
\vskip 2mm

For iterates of the exponent, the inverse function, id est, arctetration, or abelexponent, is also required. It is considered in the next chapter.

\chapter{~ Natural arctetration}
\baselineskip 18.6pt

\begin{figure}[h]
\vskip -36pt
\hskip -30pt
\sx{2.5}{\begin{picture}(220,124)
\put(0,0){\includegraphics{vladi02c}}
\put(17,114){\sx{.6}{$y$}}
\put(17, 99){\sx{.58}{$4$}}
\put(17, 89){\sx{.58}{$3$}}
\put(17, 79){\sx{.58}{$2$}}
\put(17, 69){\sx{.58}{$1$}}
\put(17, 59){\sx{.58}{$0$}}
\put(12, 49){\sx{.58}{$-1$}}
\put(12, 39){\sx{.58}{$-2$}}
\put(12, 29){\sx{.58}{$-3$}}
\put(12, 19){\sx{.58}{$-4$}}
\put(12,  9){\sx{.58}{$-5$}}
\put(178,5.2){\sx{.6}{$x$}}
\put(160,5.2){\sx{.6}{$6$}}
\put(140,5.2){\sx{.6}{$4$}}
\put(120,5.2){\sx{.6}{$2$}}
\put(100,5.2){\sx{.6}{$0$}}
\put( 76,5.2){\sx{.6}{$-2$}}
\put( 56,5.2){\sx{.6}{$-4$}}
\put( 36,5.2){\sx{.6}{$-6$}}

\put( 64,119){\sx{.5}{$u\!=\!2.2$}}
\put( 94,119){\sx{.5}{$u\!=\!2$}}
\put(118,119){\sx{.5}{$v\!=\!0.6$}}
\put(148,119){\sx{.5}{$v\!=\!0.4$}}
\put(166, 59.8){\sx{.5}{$v\!=\!0$}}
\end{picture}}
\vskip -16pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi02.jpg}
\vskip -4pt
\caption{$u\!+\!\mathrm i v=\mathrm{ate}(x\!+\!\mathrm i y)$ 
~ ~ ~ ~ \iL{vladi02c} ~ ~ ~ ~
\iL{figsexpG} }
\vskip 2pt
\end{figure}

The inverse function of tetration, id est, arctetration, or abelexponent,
is denoted with name ate; 
$\mathrm{ate}=\mathrm{tet}^{-1}$. 
Complex map of arctetration is shown in figure 
\ref{vladi02c}.\vskip 2mm

Arctetration satisfies the Abel equation
\be
\mathrm{ate}\Big(\exp(z)\Big)=\mathrm{ate}(z)+1
\label{abelex}
\eL{abelate}
and the additional condition
\be
\mathrm{ate}(1)=0
\eL{abelate10}
Properties of functions ate and the algorithm for the evaluation are described in this chapter.

\section{Evaluation of arctetration}

Arctetration can be evaluated as inverse function of tetration, using the Newton method.
Function  $\mathrm{ate}(z)$ appears as limit of sequence $g_n$ with the recurrent relation
\be
g_{n+1}=g_n+\frac
{\mathrm{tet}(g_n) - z }
{\mathrm{tet}^{\prime}(g_n)}
\eL{atenewton}
The derivative of tetration can be approximated, differentiating the approximations of tetration with elementry functions described in the previous section. The representation through the Cauchy integral \cite{analuxp} also allows the straightforward differentiation. However, in this case, several iterates by \rf{atenewton} are required to evaluate the arctetration. 
\vskip 6pt

Evaluation of arctetration through tetration using equation \rf{atenewton} is significantly slower, than evaluation of tetration.
In addition, the initial approximation $g_0$ should be specified. This specification should carry about the cutlines. In Figure \ref{vladi02c}, these cut lines are drawn parallel to the real axis. Over-vice, the recurrence by \rf{atenewton} returns a value from any of branches of the corresponding multivalued function, and the question about the range of holomorphism becomes difficult.
\vskip 2mm

In order to get efficient approximation for the arctetration ate, I deal with the corresponding Abel equation \rf{abelate}, rather than with recurrences by \rf{atenewton}. 
It worth to approximate arctetration with some function, which reproduce at least the leading terms of the asymptotic expansion of ate. This approximation is constructed below.
\vskip 6pt

Arctetration, as solution of the Abel equation \rf{abelex}, should have singularities in the fixed points of logarithm $L$ and $L^*$.
From the precious chapter, we already know, that the dominant term of the asymptotic expansion appears as the fixed point plus the appropriate exponential. This indicates, that the expansion of arctetration should begin with logarithm.
The efficient approximation of arctetration can be obtained through the expansion of function $h$ by
\be
h(z)=\mathrm{ate}(z)
- \frac{\ln(z\!-\!L)}{L}
- \frac{\ln(z\!-\!L^*)}{L^*}
\eL{atelo}
Function $h$ can be expanded to the Taylor series at unity. This expansion leads to the approximation
\be \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!
\mathrm{fsl}(z) &=& \frac{\ln(z\!-\!L)}{L} + \frac{\ln(z\!-\!L^*)}{L^*} + \sum_{n=0}^{N-1} u_n \, (z\!-\!1)^n
\iL{fsl}\\  \!\!\!\!\!\!\!\!\!\!  \!\!\!\!\!\!\!\!\!\!
\mathrm{ate}(z)&=&\mathrm{fsl}(z)+O(z\!-\!1)^N
\ee
Approximations of the first 30 coefficients of this expansion are shown in table 
\ref{vladitab2}. 
Complex map of function fsl by \rf{fsl} at $N\!=\!70$ is shown in the left hand side picture of figure \ref{vladi10} in the same notations, as in figure \ref{vladi02c}.
The central part of left map in figure \ref{vladi10} looks as a fragment from figure  \ref{vladi02c}.
\begin{table}[h]
\caption{\label{vladitab2} \large
Coefficients $u_{n}$ in expansion \rf{fsl}.
}

\sx{1.05}{\begin{tabular}{rr|rr|rr}
$n$	& $~~~~~~~~~u_{n}~~~~~~~$ &
$~~n$	& $~~~~~~~~u_{n}~~~~~~~~$ &
$~~n$	& $~~~~~~~~u_{n}~~~~~~~~$ \\
\input{vladitab2}
\end{tabular}}
\end{table}
\vskip 2mm

\newcommand \fsloax {
\put(-3,49){\sx{.45}{$y$}}
\put(-3,39){\sx{.5}{$1$}}
\put(-3,29){\sx{.5}{$0$}}
\put(-7,19){\sx{.5}{$-1$}}
\put(-7, 9){\sx{.5}{$-2$}}
\put( 7  ,5){\sx{.5}{$-\!1$}}
\put(20 , 5){\sx{.5}{$0$}}
\put(30 , 5){\sx{.5}{$1$}}
\put(40 , 5){\sx{.5}{$2$}}
\put(48 , 5){\sx{.45}{$x$}}
}
\begin{figure}[t]
\hskip 18pt
\sx{2.34}{\begin{picture}(60,55)
\put(0,0){\includegraphics{vladi10a}}
\fsloax
\end{picture}}
\sx{2.34}{\begin{picture}(60,55)
\put(0,0){\includegraphics{vladi10b}}
\fsloax
\put(32,44){\sx{.5}{$D_{\rm A}\!<\!1$}}
\put(21,33){\sx{.55}{$15$}}
\end{picture}}
\sx{2.34}{\begin{picture}(50,55)
\put(0,0){\includegraphics{vladi10c}}
\fsloax
\put(3,33){\sx{.5}{$D_{\rm B}\!<\!1$}}
\put(32,33){\sx{.55}{$15$}}
\end{picture}}
\vskip -16pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi10.jpg}
\vskip -4pt
\caption{%\large
$u\!+\!\mathrm i v\!=\!\mathrm{slo}(x\!+\!\mathrm i y)$
by \rf{fsl}, left, and the agreements by 
$D_{\mathrm{A}}\!=\!D_{\rm{A}}(x\!+\!\mathrm i y)$,
$D_{\mathrm{B}}\!=\!D_{\rm{B}}(x\!+\!\mathrm i y)$,
by formulas
\rf{sloE}, \rf{sloL}
\label{vladi10} \label{figslo}}
\vskip 8pt
\end{figure}
Range of validity of approximation  \ref{fsl} is limited. In order to show this range, the central and right hand side maps in figure \ref{vladi10} show the agreements
\be \!\!\!\!\!\!\!\!\!\!
D_{\mathrm A}(z)&=&-\lg\!\left( \frac
{|\mathrm{fsl}(\exp(z))\!-\!1-\mathrm{fsl}(z)|}
{|\mathrm{fsl}(\exp(z))\!-\!1|+|\mathrm{fsl}(z)|}
\right) ~\iL{sloE}\\  \!\!\!\!\!\!\!\!\!\!
D_{\mathrm B}(z)&=&-\lg\!\left( \frac
{|\mathrm{fsl}(\ln(z))\!+\!1-\mathrm{fsl}(z)|}
{|\mathrm{fsl}(\ln(z))\!+\!1|+|\mathrm{fsl}(z)|}
\right) ~\iL{sloL}
\ee
Inside the inner loops in the central and right hand side pictures of figure  \ref{vladi10}, the agreement is better than 14. These domains are marked with symbol ``15''.
Figure \ref{vladi10} indicates, that at 
$|z\!-\!1|\!<\!1.4$, 
the approximation $\mathrm{fsl}(z)$ provides of order of 14 significant figures.
The precision of this approximation is a little bit worse in vicinity of fixed points $L$ and $L^*$.
This could be expected: First, there, the small variation of argument leads to the significant variation of value of function.
Second, these points are at the edge of the range of convergence of expansion in equation \rf{fsl}.
\vskip 2mm

For implementation of natural arctetration, it is sufficient to prepare the efficient primary approximation for the domain
sickle, defined with
\be
\mathrm{sickle}=\{z \in \mathbb C : \Re(z)\ge L, |z|<L \}
\eL{sicle}
then, for other values of the argument, function can be expressed through the Abel equation \rf{abelate}.
The region sickle is shaded in maps of figures \ref{vladi02c}, \ref{vladi10}. This leads to implementation, denoted with
FSL. I check the self-consistency of implementation FSL with implementation FSE by \rf{FSE}. The numerical test of the relations below had been performed:
\be
\mathrm{ate}(\mathrm{tet}(z))=z  ~ ~ \iL{atetet}\\
\mathrm{tet}(\mathrm{ate}(z))=z ~ ~ \iL{tetate}
\ee
These relations are tested for the complex double implementations 
$\mathrm{tet}\!\approx\! \mathrm{FSE}$ 
and
$\mathrm{ate}\!\approx\! \mathrm{FSL}$.
Figure \ref{vladi11} shows the maps of the agreements
\be
D_{\rm at}(z)= -\ln\left( \frac
{|\mathrm{FSL}(\mathrm{FSE}(z)) - z|}
{|\mathrm{FSL}(\mathrm{FSE}(z))| + |z|}
\right)
\iL{Dat} \label{FSLFSE}
\\
D_{\rm ta}(z)= -\ln\left( \frac
{|\mathrm{FSE}(\mathrm{FSL}(z)) - z|}
{|\mathrm{FSE}(\mathrm{FSL}(z))| + |z|}
\right)
\iL{Dta}\label{FSEFSL}
\ee

\newcommand \vladiEL {
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\put(-18,544){\sx{2.3}{4}}
\put(-18,494){\sx{2.3}{4}}
\put(-18,444){\sx{2.3}{3}}
\put(-18,394){\sx{2.3}{2}}
\put(-18,344){\sx{2.3}{1}}
\put(-18,294){\sx{2.3}{0}}
\put(-36,244){\sx{2.3}{$-1$}}
\put(-36,194){\sx{2.3}{$-2$}}
\put(-36,144){\sx{2.3}{$-3$}}
\put(-36, 94){\sx{2.3}{$-4$}}
\put(-36, 44){\sx{2.3}{$-5$}}
\put( 30, -20){\sx{2.3}{$-3$}}
\put( 80, -20){\sx{2.3}{$-2$}}
\put(130, -20){\sx{2.3}{$-1$}}
\put(196, -20){\sx{2.3}{$0$}}
\put(246, -20){\sx{2.3}{$1$}}
\put(296, -20){\sx{2.3}{$2$}}
\put(346, -20){\sx{2.3}{$3$}}
\put(392, -20){\sx{2.3}{$x$}}
}

\begin{figure}[t]
\vskip 16pt
\hskip 16pt
\sx{.47}{\begin{picture}(456,580)
\put(0,0){\includegraphics{vladi11a}}
\vladiEL
\put(110,520){\sx{2.8}{$10\!<\!D\!<\!13$}}
\put(154,420){\sx{2.4}{$12\!<D\!<\!14$}}
\put(160,310){\sx{2.8}{$D\!>\!14$}}
\put(154,168){\sx{2.4}{$12\!<D\!<\!14$}}
\put(110,64){\sx{2.8}{$10\!<\!D\!<\!12$}}
\put(20, 320){\sx{2.8}{$D\!<\!1$}}
\put(310,560){\sx{2.8}{$D\!<\!1$}}
\put(310,20){\sx{2.8}{$D\!<\!1$}}
\end{picture}}
\sx{.47}{\begin{picture}(388,580)
\put(0,0){\includegraphics{vladi11b}}
\vladiEL
\put( 20,379){\sx{2.5}{$12\!<\!D\!<\!14$}}
\put( 20,210){\sx{2.5}{$12\!<\!D\!<\!14$}}
\put(200,310){\sx{3}{$D\!>\!14$}}
\end{picture}}
\vskip 6pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Vladi11.jpg}
\vskip -6pt
\caption{\large $D\!=\!D_{\rm at}(x\!+\!\mathrm i y)$,
left, and
$D\!=\!D_{\rm ta}(x\!+\!\mathrm i y)$, 
right, by 
(\ref{FSLFSE}) \EN{and}\RU{и} (\ref{FSEFSL}) 
\label{figslogsexp}
\iL{vladi11}
\label{figss}
}
\vskip 8pt
\end{figure}
\vskip 2mm

In figure \ref{vladi11}, the left hand side map shows 
$D=D_{\rm at}(x\!+\!\mathrm i y)$ and the right hand side map shows 
$D=D_{\rm ta}(x\!+\!\mathrm i y)$ in the $x$, $y$ planes.
The levels $D=\mathrm {const}$ are drawn with interval 2; as in other maps of agreement, the additional level $D=1$ is shown with thick line. This line indicates the boundary of the range of validity of relation \rf{atetet}.
\vskip 2mm

As for many other inverse functions, the range of validity to relation \rf{atetet} is limited.
Width of the strip, where the relation \rf{atetet} takes place, is determined by the asymptotic periodicity of tetration in the upper and the lower parts of the complex plane. In vicinity of the real axis, the strip becomes wider, showing some kind of 
along the real axis to infinity. Thickness of this ``beak'' 
reduces quickly at the growth of the real part of the argument.
\vskip 2mm

The numerical tests confirm, that the approximations of tetration and arctetration are self-consistent. The complex double 
implementations provide of order of 14 significant figures.
\section{About names}
\baselineskip 21pt

The numerical implementations of algorithm FSE and FSL  described above are loaded to TORI as\\
\url{https://mizugadro.mydns.jp/t/index.php/Fsexp.cin} and\\
\url{https://mizugadro.mydns.jp/t/index.php/Fslog.cin}\\
The names of these routines are discussed in this section.
\vskip 2mm

Names of function FSEXP and FSLOG are historic. They may mean ``Fast Super EXPonential'' and ``Fast Super LOGarithm.
Hernryk Trappmann even wanted to add my last name to the identifier of each of these two functions.
He had believed, that ``my tetration'' is not unique, and not so principal, as I claim, using names ``tet'' and ``ate''. Henryk wanted to use names
``tet'' and ``ate'' for the ``true'' tetration and arctetration, ``more natural'', more ``true'', than the functions I had constricted. 
\vskip 6pt 

After the long discussion and the heavy deduction, Henryk had to agree, that the functions I have constructed are unique
\cite{uniabel}, and, in this sense, true and the only true tetration and arctetration.
The routines FSEXP and FSLOG were already implemented that time, and I decided not to change the notations:
the poor sistem of notation is still better and causes less confusions, than two ``good'' systems of notations.
\vskip 2mm

I need to mention, that name FSLOG (Fast Super LOGarithm)  is even more idiotic, than FSEXP.
Name FSLOG makes impression, that it is superfunction of logarithm, while it is not really so.
Superfunction of natural logarithm van be written as
\be
z \mapsto \mathrm{tet}(-z)
\ee
I think, this function does not deserve to have a special name assigned.
\vskip 2mm

I am far from eugenic ideas to refine the human rase, nor the system of notations. The best system of notations should win the competition with other notations. All this should be considered as my excuse to keep names FSEXP and FSLOG for the approximations and the numerical implementations of tetration tet and arctetration ate.
\vskip 2mm

Many superfunctions of natural exponent can be constructed with transformation \rf{tif}, just misplacing the argument of tetration with some periodic real-holomorphic function. The range of holomorphism of these transforms is narrower, than the range of holomorphism of tetration. Now, I see no need to give them special names.
\vskip 2mm

I expect, in future, even more efficient implementations for tetration will be suggested. Then, they may be called with the same names, as the name of the functions tet and ate, in the same way, as in the algorithmic languages the implementation of sin is denoted with the same name as the function.
\vskip 2mm

After to eliminate the potential confusion with names, the tetration and arctetration can be used for the iterates of the exponent. These iterates are described in the next section.

\section{Iterates of exponent}

\baselineskip 18pt
\begin{figure}[t]
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\put(407,192.6){\sx{2.9}{$-2$}} 
\put(407,092.4){\sx{2.9}{$-3$}} 
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\put(396,408){\sx{2.9}{$0$}} 
\put(497,408){\sx{2.9}{$1$}} 
\put(597,408){\sx{2.9}{$2$}} 
\put(697,408){\sx{2.9}{$3$}} 
\put(787,408){\sx{3}{$x$}} 
 \put(6,748){\sx{3}{\rot{4}$n\!=\!3.2$\ero}} 
\put(6,708){\sx{3}{\rot{3}$n\!=\!3.1$\ero}} 
\put(6,675){\sx{3}{\rot{3}$n\!=\!3$\ero}} 
\put(5,643){\sx{3}{\rot{3}$n\!=\!2.9$\ero}} 
\put(5,497){\sx{3}{\rot{0}$n\!=\!2$\ero}} 
\put(4,397){\sx{2.9}{\rot{0}$n\!=\!1$\ero}} 
\put(6,345){\sx{3}{\rot{1}$n\!=\!0.6$\ero}} 
\put(7,308){\sx{3}{\rot{4}$n\!=\!0.4$\ero}} 
\put(7,280){\sx{3}{\rot{5}$n\!=\!0.3$\ero}} 
\put(8,244){\sx{3}{\rot{6}$n\!=\!0.2$\ero}} 
\put(9,185){\sx{3}{\rot{11}$n\!=\!0.1$\ero}} 
\put(48,34){\sx{3.1}{\rot{45}$n\!=\!0$\ero}} 
\put(202,5){\sx{3}{\rot{76}$n\!=\!-0.1$\ero}} 
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\put(299,5){\sx{3}{\rot{84}$n\!=\!-0.3$\ero}} 
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\put(724,5){\sx{3}{\rot{83}$n\!=\!-3.1$\ero}} 
\put(764,5){\sx{3}{\rot{82}$n\!=\!-3.2$\ero}} 
 
\put(484,600){\sx{3.7}{\rot{70}$y\!=\!\exp(x)$\ero}} 
\put(641,630){\sx{3.4}{\rot{44}$y\!=\!x$\ero}} 
\put(650,484){\sx{3.6}{\rot{19}$y\!=\!\ln(x)$\ero}} 
\end{picture} }
\vskip -6pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:ExpIte4T.jpg}
\vskip -8pt
\caption{\large 
$y\!=\!\exp^n(x)$ 
by \rf{expn} 
for varuous
$n$ 
\iL{expiteplot} }
\vskip 4pt
\end{figure}

Tetration tet and arctetration ate, as superfunction and abelfunction of exponent, specify, determine the non-integer iterates:
\be
\exp^n(z)=\mathrm{tet}(n+\mathrm{ate}(z))
\eL{expn}
Here, number $n$ of iterates, has no need to be integer (although, of course, can be integer too).
For real values of argument, iterates of exponent by equation \rf{expn} are shown in figure \ref{expiteplot},
$y\!=\!\exp^n(x)$. 
Lines, that correspond to integer $n$ (except $n\!=\!0$), are thick. These lines correspond to
$y\!=\!\exp(\exp(\exp(x)))$,
$y\!=\!\exp(\exp(x))$,
$y\!=\!\exp(x)$,
$y\!=\!\ln(x)$,
$y\!=\!\ln(\ln(x))$,
$y\!=\!\ln(\ln(\ln(x)))$. Higher integer iterate happen to be out of range of the figure.
\vskip 2mm

Complex maps of iterates of the exponent are collected in figure \ref{expitemap}.
\newcommand \LeLa {
\put(-24,384){\sx{4}{$y$}}
\put(-24,288){\sx{4}{$3$}}
\put(-24,188){\sx{4}{$2$}}
\put(-24,88){\sx{4}{$1$}}
\put(-24,-12){\sx{4}{$0$}}
}
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\put(270,-36){\sx{3.8}{$-1$}}
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\put(590,-36){\sx{3.8}{$2$}}
\put(690,-36){\sx{3.8}{$3$}}
\put(786,-36){\sx{3.8}{$x$}}
}
\begin{figure}%[H]
~
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\put(0,0){\ing{exp10map}} \LeLa \put(20,342){\sx{7}{$n\!=\!1$}} 
\put(46,300){\rot{0}\sx{5}{$v\!=\!0$} \ero}
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\put(474,4){\rot{56}\sx{5}{$u\!=\!2$} \ero} 
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm10map}} \put(20,340){\sx{7}{$n\!=\!-1$}} 

\put(30,144){\rot{56} \sx{5}{$u\!=\!1.4$}\ero} %%%%%%
\put(100,114){\rot{55} \sx{5}{$u\!=\!1.2$}\ero} %%%%%%
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\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp09map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.9$}}
\put(50,202){\rot{20}\sx{5}{$v\!=\!0$} \ero}
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\put(482, 10){\rot{52}\sx{5}{$u\!=\!2$} \ero}
 \end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm09map}}  \put(20,340){\sx{7}{$n\!=\!-0.9$}} 
\put(180,198){\rot{27} \sx{5}{$u\!=\!1$} \ero}
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\put(649,-13){\rot{0} \sx{5}{$v\!=\!0$}\ero} 
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp08map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.8$}}
\put(168,198){\rot{35}\sx{5}{$v\!=\!0$} \ero}
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\put(430,146){\rot{13}\sx{5}{$u\!=\!0$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
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\put(488, 1){\rot{59}\sx{5}{$u\!=\!2$} \ero}
 \end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm08map}}  \put(20,340){\sx{7}{$n\!=\!-0.8$}} 
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\put(648,30){\rot{8} \sx{5}{$v\!=\!0.2$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp05map}} \LeLa  \put(20,340){\sx{7}{$n\!=\!0.5$}} 
\put(360,294){\rot{11} \sx{5}{$u\!=\!-2$} \ero} 
\put(382,222){\rot{22} \sx{5}{$u\!=\!-1$} \ero} 
\put(420,146){\rot{38} \sx{5}{$u\!=\!0$} \ero} 
\put(490,99){\rot{54} \sx{5}{$u\!=\!1$} \ero} 
\put(550,82){\rot{58} \sx{5}{$u\!=\!2$} \ero}
\put(600,66){\rot{62} \sx{5}{$u\!=\!3$} \ero}
\put(10,122){\sx{5}{\bf cut}} 
\put(680,90){\rot{-16} \sx{5}{$v\!=\!3$} \ero}
\put(680,24){\rot{-8} \sx{5}{$v\!=\!1$} \ero}
\put(210,154){\rot{78} \sx{5}{$v\!=\!0$} \ero} 
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm05map}}  \put(20,340){\sx{7}{$n\!=\!-0.5$}}  
\put(602,386){\rot{-61} \sx{5}{$u\!=\!2$} \ero}
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\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp02map}}  \put(20,340){\sx{7}{$n\!=\!0.2$}}  \LeLa %\DoLa
\put(353,224){\rot{60} \sx{5}{$u\!=\!-1$} \ero}
\put(452,224){\rot{66} \sx{5}{$u\!=\!0$} \ero}
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\put(10,122){\sx{5}{\bf cut}}
\put(680,328){\rot{-15} \sx{5}{$v\!=\!5$} \ero}
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\put(680,194){\rot{-10} \sx{5}{$v\!=\!3$} \ero}
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\put(680,56){\rot{-3} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(200,240){\rot{-31} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm02map}}  \put(20,340){\sx{7}{$n\!=\!-0.2$}}  %\DoLa 
\put(158,312){\rot{-57} \sx{5}{$u\!=\!-1$} \ero}
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\put(10,122){\sx{5}{\bf cut}}
\put(680,268){\rot{9} \sx{5}{$v\!=\!2$} \ero}
\put(470,132){\rot{8} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\put(680,124){\rot{4} \sx{5}{$v\!=\!1$} \ero}
\put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero}
\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}\vskip 8pt

~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp01map}}  \put(20,340){\sx{7}{$n\!=\!0.1$}}  \LeLa \DoLa
\put(340,228){\rot{74} \sx{5}{$u\!=\!-1$} \ero}
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\put(680,237){\rot{-5} \sx{5}{$v\!=\!3$} \ero}
\put(680,154){\rot{-4} \sx{5}{$v\!=\!2$} \ero}
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\put(114,210){\rot{-19} \sx{4.6}{$v\!\approx\!1.337$} \ero}
\end{picture}} ~
\sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm01map}}  \put(20,340){\sx{7}{$n\!=\!-0.1$}}  \DoLa 
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\put(8,-13){\sx{5}{\bf cut}} 
\end{picture}}
\vskip 8pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Expitemap.jpg}
\vskip -2pt
\EN{\caption{Maps of iterates of natural exponent by \rf{uxexpxy}  \label{expitemap}}}
\RU{\caption{ $u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)$ по формуле \rf{uxexpxy} для различных $n$ \iL{expitemap}}}
\end{figure}

Twelve maps are shown for
\be
u\!+\!\mathrm i v = \exp^n(x\!+\!\mathrm i y)
\eL{uxexpxy}
with lines $u\!=\mathrm{const}$ and lines $v\!=\mathrm{const}$ 
in the $x$, $y$ plane for various values of the number $n$ of iterate;
this $n$ is printed with big font in the upper left corner of each map.
The maps are symmetric with respect to reflection from the abscissa axis (the only, the imaginary part of the function changes its sign).
So, the only upper part of the complex plane is presented in each map.
\vskip 2mm

Maps at the top of figure \ref{expitemap} correspond to  $n\!=\!1$ and $n\!=\!-1$; these are complex maps of exponent and of logarithm.
First of them reproduces part of figure \ref{expMap}.
The exponent is holomorphic in the whole complex plane, but the logarithm has branch point at zero and the cut along the negative part of the real axis.
\vskip 2mm

The second and following rows of the figure represent the non-integer iterates. These iterates have two additional cuts along lines $y\!=\!\pm \Im(L)$; here 
$L\!\approx\!
0.3181315
\!+\! 
1.3372357
\, \mathrm i\,$ 
is fixed point of logarithm, id est, solution of equation $L\!=\!\ln(L)$. By default, all the cut lines are directed parallel to the real axis (axis $x$ in the figure, abscissa) toward the negative direction of the real axis.
In such a way, for negative non-integer $n$, the map has 3 cut lines (and that in the lower half-plane is not seen, as it is out of field of view of the map).
\vskip 2mm

The thick lines in figure \ref{expitemap} corresponds to the integer values of $u$ or $v$. The thin lines are drawn with interval $0.2$; the additional lines $u\!=\!\Re(L)$ and $v\!=\!\Im(L)$ are also drawn. These lines always cross each other at the fixed point $L$.
\vskip 2mm

Figure  \ref{expitemap} shows the gradual transition of the map for the exponential (top of the left column) the map for the logarithm (top of the right hand side column). As the number $n$ of iterate reduces from unity to zero, the web of the lines 
$u\!=\!\mathrm{const}$ and lines $v\!=\!\mathrm{const}$ rotate around the fixed point $L$, and become uniform rectangular grid at $n\!=\!0$. At this value, the horizontal cuts along lines $\pm \Im(L)$ disappear, but they appear again, as $n$ becomes negative non-integer. At $n\!<\!0$, the additional branch point comes from $-\infty$ at the real axis and moves toward zero, as $n$ becomes minus unity. With integer $n$, the branch points $L$ and $L^*$ disappear.
\vskip 2mm

Maps of non-integer iterates can be plotted also for other transfer functions, considered in this Book. The readers are 
invited to download the implementations of the superfunctions and the abelfunctions, and plot the corresponding complex maps of the iterates.

\section{Lessons of natural tetration and arctetration}

On the base of representation of tetration through the Cauchy integral, through the solution of the integral equation
\rf{3dot2}, 
one can express also derivatives of this function;
differentiation of the integrands in the right hand side of 
\rf{3dot2}
is straightforward.
Namely in this way, the derivatives of tetration for real and for pure imaginary values of the argument had been evaluated for the tables 1 and 2 in publication 
\cite{analuxp}. In this section, I suggest some philosophic speculations about tetration and arctetration.

\vskip 2mm

Complex map of tetration $\mathrm {tet}$ is shown in figure \ref{tetmap}, and its behaviour along the real axis is shown in figure \ref{TetPlot}. 
Properties
\rf{T715}-\rf{T718}
first were observed with various approximations of tetration with elementaty functions, and then postulated.
The approximations reproduce values of tetration in vicinity of integer values of the argument,
\be
\mathrm{tet}(-2+\varepsilon) &=& \log(\varepsilon) +\mathrm{const}+O(\varepsilon) \\
\mathrm{tet}(-1)&=&0\\
\mathrm{tet}(0)&=&1\\
\mathrm{tet}(1)&=&\mathrm e\\
\mathrm{tet}(2)&=&\mathrm e^2
\ee
Then, the agreement at the substitution of the fitting function into the transfer equation
\rf{transferexp} had been minimised for complex values of the argument.
\vskip 2mm
The behaviour similar to properties 
\rf{T715}-\rf{T718}
had been detected with various fitting functions. 
Then these properties were formulated as definition of tetration, id est, just postulated.
First, I did not expect this set of postulates to be self-consistent. Contrary, I tried to find some contradiction;
I expected to use such a contradiction as a proof of non-existence of holomorphic tetration.
Such a non-existence would be an upgrade of the proof by
M.Hooshmand \cite{hooshmand}, 
that uses the doubtful assumption about monotonous behaviour of the derivative of tetration; I tried to find a proof, that does not use this assumption.
Expression of tetration through the Cauchy integral   \cite{analuxp} allows to make the precise approximations \cite{vladie},
and no internal contradictions in the assumptions  \rf{T715}-\rf{T718} had been detected. This leads to the conjecture about existence and uniqueness of tetration, that later had been confirmed with the careful analysis 
\cite{uniabel}.
I show the first primitive approximations in figure \ref{analuxpmap},  as they answer the frequent question by colleagues:
``{\em How did you guess?}''. I think this heuristic approach can be used also for analysis of other (and more complicated) functional equations.
\vskip 2mm

Tetration and arctetration significantly extend the arsenal of functions, available for the description of physical phenomena.
In particular, the non-integer iterates of exponent can be useful in description of processes, that grow faster than any polynomial, but slower than any exponent.
\vskip 2mm

Following the TORI axioms, I formulate mainly the practical problems. From the point of view of applications, not the proof by itself is important, but the strong indication, that the system of postulates is not self-contradictory. The multiple (failed) attempts to reject the conjecture of existence and uniqueness can be considered as such indication.
\vskip 2mm

Some ``pure mathematicians'' believe, that the only rigorous proof has a scientific value. In order to show, that actually it is not so,
I suggest the example with the Euclid 
axioms of planimetry. Those axioms can be deduced from the properties of the coordinate plane.
It is not so difficult, although first, one had to provide the accurate definitions of sin and cos as solution of the corresponding system of differential equations, check that their properties lead to the Pythagoras theorem and other properties, known as the Euclid Axioms.
In the elementary school, however, till now, the teachers begin with the postulating the Euclid axioms.
I believe, the superfunctions should become a pretty elementary tool, and their properties (including those of tetration) could be just postulated - in the similar way, as the Euclid axioms. If someone wants to reduce the amount of axioms, one may begin wight he Euclud axioms, having no need to deal with tetration and other superfunctions. I hope, one day, the beautiful, short, simple and rigorous proof of the existence and uniqueness will be formulated.
\vskip 2mm
Form my side, I make all possible efforts in order to simplify refutation of my concept (for the case, if one day someone will be able to refute them). I load the figures from this book to my site as \\
\url{https://mizugadro.mydns.jp/t/index.php/Category:BookPlot} \\
\url{https://mizugadro.mydns.jp/t/index.php/Category:BookMap}\\ 
and I supply them with generators in C++ and Latex. 
Everybody can reproduce the figures, and plot the new figures,
trying to find a hint to any internal contradictions in the concepts suggested.
Of course, any other alternative hypothesis can be considered too,
as it is shown in figure \ref{sheep}.
\vskip 2mm

\vskip 2mm
\begin{figure}[h]
\begin{center}
\url{https://mizugadro.mydns.jp/t/index.php/File:BlackSheep.png}
\vskip 18pt
\sx{.8}{\begin{picture}(454,350)
\put(0,0){\ing{blacksheep}}
\put(-2,-2){\color{white} \rule{8pt}{356pt}}
\put(-2,-2){\color{white} \rule{458pt}{8pt}}
\put(456,-2){\color{white} \rule{8pt}{356pt}}
\put(-2,346){\color{white} \rule{458pt}{8pt}}
\end{picture}}\end{center}
\vskip -24pt
\caption{\large \baselineskip 18pt 
Two mathematicians go to the First International Congress on Superfunctions,
and discuss the color of the ship they see from the train: 
{\em Your assumption, dear colleague, seems to be not obvious, it is not supported with observations. 
Yet, all what we can conclude, that there is at least one sheep in this country,
and at least the right hand side of this sheep is black.}
\iL{sheep}}
\vskip 12pt
\end{figure}

After to see, how the natural tetration comes from the Cauchy integral in a pretty natural way,
I had constructed similar maps for other values of base $b$, namely, for $b\!=\!10$, $b\!=\!2$ and
$b\!=\!1.5$,  
but I did not revealed any new property, that could be difficult to expect,
looking at the natural tetration.
The most of curves in figure \ref{tet10bx}, considered later,
can be plotted with the Cauchy integral by very similar algorithms.
I was sure, that my mission about tetration is finished.
Then, Henryk Trappmann wanted still to reduce $b$;
he asked me, Wether I can evaluate in the similar way tetration to base
$b=\sqrt{2}$.
I had to confess, that I cannot. But I told, that I can do it by another way \cite{sqrt2}.
That ``another way'' happened to be even simpler, than application of the Cauchi integral; so, I described it in the previous chapters as ``regular iteration''.
However, namely tetration to base $b\!=\!\sqrt{2}$ is not described above;  this tetration is matter
of the next chapter.

\chapter{~ Tetration to base $b\!=\!\sqrt{2}$}
\baselineskip 20.4pt

\begin{figure}[h]
\vskip -16pt
\hskip 66pt
\sx{.5}{\begin{picture}(610,620)
\put(0,0){\ing{sqrt27a}}
\put(172,600){\sx{4}{$2$}}
\put(172,500){\sx{4}{$1$}}
\put(140,298){\sx{4}{$-1$}}
\put(140,198){\sx{4}{$-2$}}
\put(140, 98){\sx{4}{$-3$}}
\put( -9,366){\sx{4}{$-2$}}
\put( 89,366){\sx{4}{$-1$}}
\put(300,366){\sx{4}{$1$}}
\put(401,366){\sx{4}{$2$}}
\put(502,366){\sx{4}{$3$}}
\put(594,370){\sx{4.2}{$x$}}
\put(340,530){\sx{4.3}{\rot{5}$y\!=\!\mathrm{tet}_{\sqrt{2}}(x)$\ero}} 
\put(380,200){\sx{4.2}{\rot{-45}$y\!=\!-x$\ero}} 
\end{picture}} 
\vskip -4pt
\hskip 50pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt27t.jpg}
\vskip -4pt
\caption{ \large
$y=\mathrm{tet}_{\sqrt{2}}(x)$ 
\label{figreal}
\label{figJJ}
\label{figjj}
\iL{sqrt27a}
}
\vskip -4pt
\end{figure}

In chapter 13 above, the natural tetration is constructed and evaluated.
I mean, tetration to base
$b\!=\!\mathrm e \!\approx\! 2.71~$.
For other bases, the definition of tetration should be generalised.
This generalisation is suggested in this chapter.
I try to follow the principles ``from simple to complicated'' and ``from specific to general''.
First, I consider the specific base $b\!=\!\sqrt{2}$. 
For this base, the graphic of tetration is shown in figure 
\ref{sqrt27a}. 
Namely for this base, the graphic looks especially symmetric.
Below I show, that this is just visual impression, and the apparent symmetry $x\leftrightarrow -y$ is only approximation.
\section{Definition}
\baselineskip 18.8pt

Tetration to the real base $b\!>\!1$ is real-holomorhpic function  $f\!=\!\mathrm{tet_b}$, that satisfies the transfer equation 
\be
f(z\!+\!1)=b^{f(z)}
\eL{sqrt2transfer}
at least for $\Re(z)>-2$,
and is bounded at least in the range
$|\Im(z)| \le 1$, and, in addition, the specific (the same for all $b$) value at zero is assumed:
\be
f(0) &=& 1     ~ ~ ~ ~ ~ ~ ~ ~\iL{sqrt2f01}
\ee
\vskip 2mm
Here,
function $T\!=\!\exp_b$ appears as the transfer function, and tetration $f$ as its superfunction.
\vskip 2mm

The Reader is invited to check, that the natural tetration
$\mathrm{tet}\!=\!\mathrm{tet}_{\mathrm e}$,
considered in chapter 14, also falls into into this definition.
Below, the tetration to various bases is considered.
In particular, this chapter deals with the special case
$b\!=\!\sqrt{2}$.
Exponential to this base is shown in figures
\ref{exq2plo} and \ref{ExpQ2map}.
Namely for this case, in section 9, the growing super exponential 
$\mathrm{SuExp}_{\sqrt{2},5}$,  
is constructed; graphic $y\!=\!\mathrm{SuExp}_{\sqrt{2},5}$ is shown in figure 
\ref{sqrt23uplot}. %9.4.
\vskip 2mm

Here, for the same base $b\!=\!\sqrt{2}$, id est, for the same transfer function, I describe another superexponent,
namely, tetration. Its explicit plot is shown in figure \ref{sqrt27a} 
and it is pretty different from the plot of the growing superexponent
$F\!=\!\mathrm{SuExp}_{\sqrt{2},5}$ shown in figure  
\ref{sqrt23uplot}.

\section{Again regular iteration }
In this section, again I use the regular iteration, as in chapter 9, in other to construct another superfunction, namely, tetration,
for the exponential to base $\sqrt{2}$ as the transfer function.
This construction appears as an example to evaluation of tetration
$\mathrm{tet}_b$ for $1\! < \!b\! < \! \exp(1/\mathrm e)$. 
\vskip 6pt

In such a way, here I consider the case
$b\!=\!\sqrt{2}$. 
Explicit plot of this tetration is shown in figure 
\ref{sqrt27a}.
The construction below is quite analogous to the construction of the growing super exponent to the same base,
presented in chapter 9. Some formulas in this section are taken from publication
\cite{sqrt2}.
\vskip 2mm

The fixed points of exponent to base $b\!=\!\sqrt{2}$ 
are considered in chapter 9, see figure \ref{ExpQ2plot}. These fixed points are 2 and 4. 
In that chapter, the superexponent $F_5=\mathrm{SuExp}_{\sqrt{2},5}$ is built at the fixed point $L\!=\!4$.
Along the real axis, that superexponential grows monotonously from 4 to infinity.
For that superexponent, the equation $F_5(z)\!=\!1$ has no real solution; that superexponent is not tetration.
In order to get tetration, we should use the fixed point $L\!=\!2$. This case is considered in this chapter.
The corresponding superfunction  $\Phi$ can be expanded as follows:
\be
f(z)=2+\varepsilon+\sum_{m=2}^{M-1} v_m \varepsilon^m 
\eL{sqrt2fas}
\be
\Phi(z)=f(z)+O(\varepsilon^n)
\eL{sqrt2Phi}
where
\be %\varepsilon
\ve=\exp(kz)
\eL{sqrt2ve}
while increment $k$ and coefficients $v$ are constants.
Substitution of the asymptotic expansion  $F\!=\!\Phi$ in to the transfer equaiton
\be
F(z\!+\!1)=\exp\Big(\ln\Big(\sqrt{2}\Big) \, F(z)\Big)
\eL{sqrt2transfereq}
determines the increment
\be
k=\ln(\ln(2))\approx -0.3665129205816643
\eL{sqrt2k}
and leads to the chain of equations for coefficients $v$. I set $v_1\!=\!1$; then, 
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\begin{array}{rcl}
v_{2}=&\frac{\ln(2)/4}{\ln(2)-1} &\approx -0.56472283831773236365\\
v_{3}=&\!\!\frac{\ln(2)^{2}(2+\ln(2))/24} {(\ln(2)-1)(\ln(2)^2-1)} \!\! &\approx \phantom{-}0.33817758685118329988 
\end{array}
\eL{sqrt2v2}
Approximate values of coefficients $v$ are collected in table 
\ref{sqrt2tablevV}. 
\vskip 2mm

At fixed number $M$ of terms in the right hand side of equation \rf{sqrt2fas}, function $f$ and be considered as approximation
of superfunction with certain asymptotics, namely, that grows at infinity, approaching the fixed point 2. 
This superfunction appears as limit
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\Phi(z)=\lim_{n \rightarrow \infty} T^{-n}(f(z\!+\!n))=\lim_{n \rightarrow \infty} \log_b^{\, n}(f(z\!+\!n))
\eL{sqrt2F}
does not depend on the number $M$ of terms in the right hand side of equation \rf{sqrt2fas}. 
However, at large $M$, the limit in the right hand side of \rf{sqrt2F} converges faster.
\vskip 2mm

Tetration to base $b\!=\!\sqrt{2}$ that satisfies condition  \rf{sqrt2f01}, appears as function $\Phi$ with displaced argument,
\be
\mathrm{tet}_{\sqrt{2}}(z)=\Phi(x_1\!+\!z)
\eL{sqrt2tetF}
where
$x_1 \!\approx\! 1.25155147882219$ 
is solution of equation
$\Phi(x_1)\!=\!1$. 
The readers are invited to verify, that this tetration satisfies  the conditions, formulated in the section 1 of this chapter.
For real values of the argument, graphic of this function is shown in figure \ref{sqrt27a}.
The complex map of tetration to base $b\!=\!\sqrt{2}$ is shown in figure \ref{sqrt2tetmap}.
\vskip 2mm

\begin{table}[t]
\caption{Approximations of coefficients $v$ and $V$ in expansios \rf{sqrt2fas}, \rf{sqrt2ateG} \label{sqrt2tablevV}}
\begin{center} \vskip -4pt
\begin{tabular}{rrr}
$n$ & $v_{n}\pS\pS\pS$ & $V_{n}\pS\pS\pS$ \\
\input{sqrt2table1}
\end{tabular}
\end{center}
\end{table}

\begin{figure}[h]
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\put(176,21.2){$v\!=\!0$}
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\put(65,68){$v\!=\!-1$}
\put(110,161){\rot{65}$u\!=\!2.4$\ero}
\put(123,158){\rot{49}$u\!=\!2.2$\ero}
\put(122,114){\rot{40}$v\!=\!0.2$\ero}
\put(109,84){\rot{36}$u\!=\!1.6$\ero}
\put(120,80){\rot{30}$u\!=\!1.8$\ero}
\put(88,43){\rot{27}$v\!=\!-0.4$\ero}
\put(98,32){\rot{22}$v\!=\!-0.2$\ero}
\end{picture}}
\vskip -2pt
\hskip 24pt \url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2tetmap.jpg}
\vskip -2pt
\caption{ $u\!+\!\mathrm i v= \mathrm{tet}_{\sqrt{2}}(x\!+\!\mathrm i y)$
\iL{sqrt2tetmap}}
\end{figure}

Tetration
by \rf{sqrt2tetF}, 
is periodic; the period $P$ is pure imaginary,
\be \!\!\!\!\!\!\!\!\!\!\!
P=P(\mathrm{tet}_{\sqrt{2}})= 
-\frac{2 \pi \mathrm i}{\ln^2(2)}=
-\frac{2 \pi \mathrm i}{\ln(\ln(2)) } \approx  17.14314817935485 \, \mathrm i 
\label{sqrt2T}
\label{sqrt2tetP}
\ee
I remind, the double logarithm $\ln^2(2)\!=\!\ln(\ln(2))$, but does not mean $\ln(2)^2$, 
according to notations declared at the beginning of this Book.
\vskip 2mm

As it is claimed above, tetration
$\mathrm{tet}_{\sqrt{2}}(z)$ 
is holomorphic in the strip
$|\Re(z)|\!\le\! 1$. 
The range of holomorphism is much wider than this strip.
Tetration to base $\sqrt{2}$ is holomorphic in the whole complex plane,
except the countable set of branch points and the cut lines
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\{ z\in \mathbb C:  \Re(z)\le 2, ~ \Im(z)\!=\!n \Im(P) ~,~  n\in \mathbb N \}
~ ~ ~ ~
\eL{sqrt2tetCuts}
Outside these cuts, tetration approaches the fixed points of the corresponding logarithm, to 2 or to 4,
at the increase or decrease of the real part of the argument, respectively.
For any real
$y$,
\be
\lim_{x \rightarrow +\infty} \mathrm{tet}_{\sqrt{2}}(x\!+\!\mathrm i y)&=&2
\eL{sqrt2tetLim1}
and for
$y\ne \Im(T) n$, $n\in \mathbb N$,
\be
\lim_{x \rightarrow -\infty} \mathrm{tet}_{\sqrt{2}}(x\!+\!\mathrm i y)&=&4
\eL{sqrt2tetLim2}
\vskip 2mm

For real $x$, function $\mathrm{tet}_{\sqrt{2}}(x)$ is defined at $x\!>\!-2$. In point  $-2$, 
the function has logarithmic singularity, as tetrations to other values of base.
The function grows from $-\infty$ at $-2$, passes through points 
 $(-1,0)$ и $(0,1)$, and continue to grow, 
approaching its limit value 2 at
$+\infty$.
\vskip 2mm

There are several reasons, why tetration to base
$b\!=\!\sqrt{2}$ 
is especially interesting.
Namely for this base $b\!=\!\sqrt{2}$, both real fixed points of $\log_b$ have integer values.
In addition, namely for this base, the graphic of function in figure 
\ref{sqrt27a} 
seems to be symmetric with respect to reflection 
$x \leftrightarrow -y$. 
In order to stress this illusion, the additional line $y\!=\!-x$ is drawn in figure \ref{sqrt27a}.
For analysis of the illusion mentioned, the inverse function should be constructed; I mean, arctetration 
$\mathrm{ate}_{\sqrt{2}}\!=\!\mathrm{tet}_{\sqrt{2}}^{~ -1}$.
The complex map of this arctetration is shown in figure \ref{sqrt2atemap}. This function is described in the next section.

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\put(151,51){\rot{-43}$u\!=\!-3$\ero}
\put(175,51){\rot{-48}$v\!=\!0$\ero}
\end{picture}}
\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2atemap.jpg}
\vskip -2pt
\caption{$ u\!+\!\mathrm i v=\mathrm{ate}_{\sqrt{2}}(x\!+\!\mathrm i y)$ \iL{sqrt2atemap}}
\end{figure}

\section{Arctetration to base $b\!=\!\sqrt{2}$}
\baselineskip 18.1pt

For the inverse function of tetration, I use name arctetration, in analogy with arcsin, arccos and arcBessel;
$\mathrm{ate}_b=\mathrm{tet}_b^{-1}$. 
Arctetration is abelfunction of exponent and satisfies the Abel equation. For base $b\!=\!\sqrt{2}$, this equation can be written as follows:
\be
G\Big( \Big(\sqrt{2}\Big)^z\Big)=G(z)+1
\eL{sqrt2abeleq}
One of solutions of equation \rf{sqrt2abeleq} is arctetration
$G\!=\!\mathrm{ate}_{\sqrt{2}}$. 
Complex map of arctetration is shown in figure
\ref{sqrt2atemap}. 
\vskip 2mm

As other abelfunctions, arctetration to base $\sqrt{2}$ can be evaluated through its asymptotic expansion,
using the Abel equation \rf{sqrt2atemap} to bring the argument into the range, where the fruncated expansion
provides the required precision.
The asymptotic expansion for the arctetration can be obtained with the asymptotic expansion of tetration.
Also, the same expansion can be obtained directly from the Abel equation  \rf{sqrt2atemap};
we should add the constant to the solution in order to satisfy condition
$\mathrm{ate}_{\sqrt{2}}(1)\!=\!0$.
\vskip 2mm

The two methods mentioned above lead to the same expansion
\be
g(z)
= \frac{1}{k} \ln\!\left(
\sum_{n=1}^{M} V_{n}\cdot (z\!-\!2)^{n}
\right)
\label{tq2L}
\eL{sqrt2ateG}
where $V$ are constant coefficients. Constant $k=\ln^2(2)$ 
is the same, as in the expansion \rf{sqrt2fas}-\rf{sqrt2ve}, see equation \rf{sqrt2k}.
Then, asymptotic solution $G$ of the Abel equation can be written as follows:
\be
G(z)=g(z)+O(z\!-\!2)^{M+1}
\eL{sqrt2GO}
Substitution of this expansion into the Abel equation  \rf{sqrt2abeleq} leads to the chain of equations for coefficients  $V$;
in particular, 
\newcommand \nn {\ln(2)}
\be
\!\!\!\!\!\!\!\!\!\! V_{1}=& 1\\
\!\!\!\!\!\!\!\!\!\! V_{2}=& -v_2= \frac{1}{4} \frac{\ln(2)}{1-\ln(2)} &\approx 0.56472283831773236365 \\
\!\!\!\!\!\!\!\!\!\! V_{3}=&  \frac{\ln(2)^2}{24} \frac {1+2 \nn} {(1-\nn)^{2}(1+\nn)} &\approx 0.29964618138408807683 
\ee 
Approximations of coefficients $V$ are collected in the second (and last) column of table \ref{sqrt2tablevV}.
\vskip 2mm

With asymptotic expansion \rf{sqrt2ateG}, the solution $G$ of the Abel equation   \rf{sqrt2abeleq} can be written as limit
\be\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
G(z)=\lim_{n\rightarrow \infty} 
g\!\left(  \exp_{\sqrt{2}}^{~  ~ n}(z) \right) -n
\eL{sqrt2Glim}
Here, in the argument of function $g$, function $\exp_{\sqrt{2}}$ is iterated $n$ times.
\vskip 2mm

In order to make arctetration $\mathrm{ate}_{\sqrt{2}}$, not only the asymptotic properties should be taken into account, but also the value at unity. So, I define tetration as
\be \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\! \!\!\!\!
\mathrm{ate}_{\sqrt{2}}(z)=G(z) \!-\! G(1)\approx G(z) \!-\! 1.25155147882219
\label{sqrt2GG1}
\eL{q2G1}
that leads to the correct value
$\mathrm{ate}_{\sqrt{2}}(1)=0$. 
In previous publication
\cite{sqrt2}, 
This arctetration is denoted with symbol
${F_{2,1}}^{-1}$; 
constant $2$ in the subscript indicates the fixed point of the transfer function, at which the regular iteration is constructed, 
and content $1$ in the subscript indicates its value at zero.
As usually, the upper index indicates the number of iterate.
\vskip 2mm

In figure \ref{sqrt2atemap}, at the complex map of arctetration, its periodicity is seen. 
This periodicity follows from the representation of arctetration through limit in equation  \ref{sqrt2atemap}.
The period $P$ is determined by the period of the exponent to base $\sqrt{2}$; 
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
P\!=\!P(\mathrm{ate}_{\sqrt{2}})\!=\!\frac{4 \pi \mathrm i}{\ln(2)} \!\approx 18.129440567308775239 \, \mathrm i
\eL{sqrt2ateP}
Imaginary part of this period is slightly greater, than that for tetration to the same base, see equation \rf{sqrt2tetP}.

\vskip 6pt
In figure  \ref{sqrt2atemap}, the isolines are reproduced at the translations along the ordinate axis for $\Im(P)$.
In addition, due to the real-holomorphism, the maps of tetration and arctetration are symmetric with respect to reflection from the real axis, id est, with respect to the up side down flip.
\vskip 2mm

In wide range of values of $z$, the identity
\be
\mathrm{tet}_{\sqrt{2}}(\mathrm{ate}_{\sqrt{2}}(z))=z
\eL{sqrt2tetatez}
is valid. This range is shaded in figure
\ref{sqrt2tetate}. %\ref{tetate}
Technically, the shading is realised as complex map of the left hand side of equation \rf{sqrt2tetatez}, treated as function of $z$ and plotted in coordinates $x\!=\!\Re(z)$ and $y\!=\!\Im(z)$.
While the equation  \rf{sqrt2tetatez} holds, the levels of the constant real part and levels of the imaginary part are parallel to the coordinate axes and form the uniform rectangular grid, that at the poor resolution looks as shading.
However, the relation \rf{sqrt2tetatez} cannot hold in a strip wider than the period of function $\mathrm{ate}_{\sqrt{2}}$;  so, the upper part and the lower part of the domain of the map are not shaded in this way.
In addition, the range of validity of equation  \rf{sqrt2tetatez} is limited at the right hand side with levels 
\be
\Im\Big( \exp_{\sqrt{2}}^n (x\!+\!\mathrm i y)\Big)=\pm \frac{|P|}{2}=\pm \frac{2\, \pi}{\ln(2)}
\eL{sqrt2en}
drawn above the map in figure \ref{sqrt2tetate}
for integer $n=0,1,2,3,4$.
In the mentioned right hand side of the figure, 
relation \rf{sqrt2tetatez} also is not valid.
\vskip 2mm

\begin{figure}[t]
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\put(26,107.4){$v\!=\!0$} 
\put(176,108){\bf cut}
\put(115,57.4){\rot{90}$u\!=\!0$\ero} 
\end{picture}}
\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2tetatemap.jpg}
\vskip -4pt
\caption{ $u\!+\!\mathrm i v=\mathrm{tet}_{\sqrt{2}}(\mathrm{ate}_{\sqrt{2}}(x\!+\!\mathrm i y))$ \iL{sqrt2tetate}}
\vskip 2pt
\end{figure}

Figure \ref{sqrt2tetate} can be considered as verification, validation, test of implementation of arctetration to base $\sqrt{2}$.
These properties and the implementations of tetration and arctetration allow to analyse approximate the symmetry $y\!=\!-x$ of 
graphic $y=\mathrm{tet}_{\sqrt{2}}(x)$ shown in figure \ref{sqrt27a}. 
\vskip 6pt

The apparent symmetry of the plot in figure \ref{sqrt27a} had been declared 
in the preamble of this chapter. The consideration had been postponed until tetration and arctetration to base $\sqrt{2}$ are described.
The approximate symmetry mentioned means, that for  $x>-2$, 
\be
\mathrm{tet}_{\sqrt{2}}(x) &\approx& - \mathrm{ate}_{\sqrt{2}}(-x) ~ ~ ~
\eL{sqrt2approx}

Properties of tetration and arctetration to base $\sqrt{2}$ indicate, that the exact equality in \rf{sqrt2approx} can take only at the set of measure zero, due to very simple and pretty fundamental reason:
tetration in the left hand side and atctetration in the right hand side of \rf{sqrt2approx} have different (incompatible) periods.
\vskip 6pt

Period of function in the left hand side of equation \rf{sqrt2approx}, see equation \rf{sqrt2tetP}, is 
$P\!\approx\! 17.143 \, \mathrm i$, 
while period in the right hand side of equation \rf{sqrt2approx}, see equation \rf{sqrt2ateP}, is 
$P\!\approx\! 18.129 \, \mathrm i$.
The different periods indicates, that these are different functions.
If two holomorphic functions coincide at the segment of finite length, they should coincide in the whole range of holomorphism.
Hence, there is no exact equality in  \rf{sqrt2approx},  for the most of $z$, nor for the exact symmetry in figure \ref{sqrt27a}.
\vskip 6pt

Deviation from the exact symmetry %in figure  \ref{sqrt2ansus}
can be characterised with function
\be
\mathrm{devia}(x)=\mathrm{tet}_{\sqrt{2}}(x) + \mathrm{ate}_{\sqrt{2}}(-x)
\eL{sqrt2simdevi}
It is shown in figure \ref{sqrt2ansus} with dashed line. 
Where the symmetry be exact, the dashed curve should follow the abscissa axis.
\begin{figure}[h]
\sx{4.1}{\begin{picture}(100,61.0)
\put(0,0){\ing{sqrt27b}}
\put(17,58.6){\sx{.36}{$y$}}
\put(46,48){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(x)+\mathrm{ate}_{\sqrt{2}}(\!-x)$}}
\put(47, 8){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{ate}_{\sqrt{2}}(x))+x$}}
\put(12.3,50.0){\sx{.33}{$0.01$}}
\put(10.0,30.0){\sx{.33}{$-0.01$}}
\put(10.0,20.0){\sx{.33}{$-0.02$}}
\put(10.0,10.0){\sx{.33}{$-0.03$}}
\put( -1,37.4){\sx{.33}{$-2$}}
\put( 9.0,37.4){\sx{.33}{$-1$}}
\put(30.2,37.4){\sx{.33}{$1$}}
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\put(100.,37.6){\sx{.36}{$x$}}
\end{picture}}
\vskip -2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt27u.png}
\vskip -8pt
\caption{%\large 
Precision of ``symmetry'' of figure \ref{sqrt27a}:
$y\!=\!\mathrm{devia}(x)$
by
\rf{sqrt2simdevi}, dashed,
and 
$y\!=\!\mathrm{devib}(x)$ by \rf{q2tetatem},
 solid
\iL{sqrt27b}
\label{sqrt2ansus}
} 
\vskip 8pt
\end{figure}

Also, the deviation is characterized with function
\be
\mathrm{devib}(x)=\mathrm{tet}_{\sqrt{2}}(-\mathrm{tet}_{\sqrt{2}}(x))+x
\eL{q2tetatem}
This dependence is shown in figure  \ref{sqrt2ansus} with solid line. Expression $-\mathrm{tet}_{\sqrt{2}}(x)$ approximate function $\mathrm{ate}_{\sqrt{2}}(-x)$.
Again, at the exact symmetry, the solid line would be just abscissa axis. In such a way, figure \ref{sqrt2ansus} indicates the range of validity of the statement about the symmetry: it reproduce of order of 2 significant figures of tetration or arctetration to base $\sqrt{2}$; however, at the segment from $-1$ to $0$ in figure \ref{sqrt27b},
the ``symmetry'' holds with 4 signifivant figures; this is pretty sufficient to cause the illusion of symmetry in figure \ref{sqrt27a}.
\vskip 2mm

Similarity of dependences  $y\!=\!\mathrm{tet}_{\sqrt{2}}(x)$ and $y\!=\!-\mathrm{ate}_{\sqrt{2}}(-x)$ for real $x$ may look occasional.
However, on the other hand, it is unavoidable for the following reasons. Every tetration to base $b\!>\!1$ has logarithmic singularity at 
point $-2$; the graphics approach vertical line $x\!=\!-2$. Graphics of all these tetrations pass through points  $(-1,0)$ and $(0,-1)$, which correspond to the symmetry discussed. In addition, for $b\!<\!\exp(1/\mathrm e)$, all the graphics have the horizontal asymptotic for large values of the augment, they approach some positive quantity (which is fixed point of logarithm). For some value of base, this quantity is $2$, that corresponds to the apparent symmetry. This value of base is just $b\!=\!\sqrt{2}$, this base is chosen as an example in this chapter as illustration behaviour of tetration and arctetration to base $b$ at $1\!<\!b\!<\!\exp(1/\mathrm e)$.
\vskip 2mm
In years 2009-2010, the apparent symmetry of graphic in figure \ref{sqrt27a} caused hard discussion.
The opponents had claimed, that the symmetry is obvious and does not require any verification.
(Before, I had observed so strong believe in the wrong and absurd statements only in the USSR; Soviet veterans had insisted on concepts of sovetism, being unable to see internal contradictions of it.)
To convince the opponents, Henryk and I had elaborated two independent demonstrations, that the exact symmetry cannot take place. %without using of properties of these functions in the complex plane.
Both these proofs are presented in publication \cite{sqrt2}.
\vskip 2mm

Readers are invited to invent some real-holomorphic function with graphic that passes through points
$(-1,0)$ 
and 
$(-0,1)$, and exponentially  approach to the vertical line $x\!=\!-2$ and horizontal line $y\!=\!2$.
I suspect, such a function will be pretty similar to tetration to base $\sqrt{2}$.
\vskip 2mm

With the suggestion above, I finish the description of arctetration to base $\sqrt{2}$.
At lest in some vicinity of the half-line $z\!<\!2$, relation \rf{sqrt2tetatez} is valid, and the pair (tetration,arctetration) can be used to iterate the exponent. These iterates are considered in the next section.

\section{Again iterate exponent to base $\sqrt{2}$}

The real-holomorphic Iterates of exponent to base $\sqrt{2}$ are considered above, in Chapter 9, for large values of positive part of the argument, with functions 
$\mathrm{SuExp}_{\sqrt{2},5}$ and 
$\mathrm{AuExp}_{\sqrt{2},5}$. 
Those iterates are presented first (and shown in figure \ref{iterEq2plot}), 
because they look similar to iterates of other growing functions, considered in the first half of this Book.
However, tetration and arctetration, constructed in this chapter, also can used to iterate the exponent to base $\sqrt{2}$.
In this section, I show, that these iterates look similar in vicinity of the interval $(2,4)$; but far from this interval, the deviation becomes strong.
\vskip 2mm

The ``regular iteration'' described above, allows to iterate the function, and iterates are regular in vicinity of the fixed point of the transfer function, used to construct the superfunction and the abelfunction.
Bot these iterates may be not regular (have singularity, branch point) at the other fixed points of the same transfer function.
Below, the illustration of this statement is presented.\vskip 2mm

Iterates of exponent to base $\sqrt{2}$ constructed with the infinitely growing superfunction
$\mathrm{SuExp}_{\sqrt{2},4}$ by \rf{q2Fz4tilde20},\rf{sqrt2regi45},\rf{Ff45} are shown in figure
\ref{iterEq2plot}.
Similar iterates can be constructed also with tetration, described in this chapter,
\be
\exp_{\sqrt{2},\mathrm d}^{~ ~ n}(z)= \mathrm{tet}_{\sqrt{2}}\left( n+ \mathrm{ate}_{\sqrt{2}}(z)\right)
\eL{sqrt2exptet}
Here, symbol ``,d'' in the subscript indicates, that the lower, ``down'' fixed point of the transfer function is used for the asymptotic of the superfunction. 
\vskip 2mm

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\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2eitet.jpg}
\vskip -2pt
\caption{ $y\!=\! \exp_{\sqrt{2},\mathrm u}^{~ n}(x)$ 
for various
$n$
\iL{sqrt2eitet}}
\end{figure}

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\vskip -4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2q2map600.jpg}
\vskip -4pt
\caption{ %Слева: $y\!=\! \exp_{\sqrt{2},2}^{~ n}(x)$ для различных $n$; 
$u\!+\!\mathrm i v\!=\! \exp_{\sqrt{2},\mathrm d}^{~ 1/2}(x\!+\!\mathrm i y)$ \iL{sqrt2q2map}}
\end{figure}

For real values of argument, iterates $\exp_{\sqrt{2},\mathrm d}^{ ~ n}$  by 
\rf{sqrt2exptet} 
are shown in figure \ref{sqrt2eitet} for various real values of $n$.
This figure is analogy of figure  \ref{iterEq2plot}, that represents the similar iterates built up with the 
infinitely growing superexponent $\mathrm{SuExp}_{\sqrt{2},5}$ and corresponding abelexponent
 $\mathrm{AuExp}_{\sqrt{2},5}$.
\vskip 3mm

Graphics in figures \ref{sqrt2eitet} and \ref{iterEq2plot} look similar.
The thick curves, for the integer iterates are, indeed, the same.
However, for the non-integer $n$, the iterates also look similar, the curves in figure \ref{sqrt2eitet} seem to be just extension, continuation of those in figure  \ref{iterEq2plot}. In the intermediate range, $2\!<\!x\!<\!4$, visually, the iterates 
$\exp_{\sqrt{2},\mathrm{d}}(x)$, 
evaluated through the tetration 
$\mathrm{tet}_{\sqrt{2}}$ and arctetration
$\mathrm{ate}_{\sqrt{2}}$, seems to be the same, as iterates
$\exp_{\sqrt{2},\mathrm{u}}(x)$, 
evaluated through the superexponent
$\mathrm{SuExp}_{\sqrt{2}}$ and abelexponent
$\mathrm{AuExp}_{\sqrt{2}}$.
Then I saw this coincidence first time, it looked strange, counter-intuitive and therefore interesting.
The matter is, that two different holomorphic functions cannot coincide at the interval of finite length.
If they are identical at the part of the real axis from 2 to 4, then they must coincide in the whole connected range of holomorphism.
\vskip 2mm

As an example, I consider the case $n\!=\!1/2$; id est, the iterates of exponent number half.
The iterate constructed with growing exponent, id est, $\exp_{\sqrt{2},\mathrm{u}}^{~1/2}$
had been shown earlier in figure \ref{iterEq2map}.
The iterate $\exp_{\sqrt{2},\mathrm{d}}^{~1/2}$, constructed with tetration and arctetration, 
is shown in figure 
\ref{sqrt2q2map}.
These two maps are not the same.
The second of them is periodic (with period $4 \pi \mathrm i /\ln(2)\!\approx\! 18.12944 \, \mathrm i$), 
while the first one is not.
These two maps look similar only in vicinity of the interval (2,4) at the real axis.
\vskip 2mm


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\vskip -6pt
\hskip 40pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2srav.png}
\vskip -4pt
\caption{
Comparison of the half iterates to base 
$\sqrt{2}$, 
constructed at fixed point 2 (dashed) and at fixed point 2 (solid curve).
In the interval between these two points, the difference
\rf{sqrt2D}
is shown, scaled with factor
$10^{24}$. \iL{sqrt2srav}}
\vskip 4pt
\end{figure}
\vskip 2mm

I felt myself confused about the identical behaviours of the half iterates along the interval $(2,4)$
of functions, that have different behaviour in the complex plane.
I thought, that I made an error implementing these functions. 
I even considered the absolutely fantastic hypothesis that I see the traces of the
{\em Mizugadro number}
\footnote{ \url{https://mizugadro.mydns.jp/t/index.php/Mizugadro_number} \\  
\url{https://budclub.ru/k/kuznecow_d_j/mizugade.shtml} Mizugadro's number (2010-2011)}, that reveals the internal contraction in the system of postulates of arithmetics
(that is used in mathematical analysis and, in particular, in the theory of holomorphic functions).
I had prepared the explicit plot the half iterates of the exponent to base $\sqrt{2}$, evaluated through the 
tetration and that evaluated through the super exponential $\mathrm{SuEx}_{\sqrt{2}}$; this plot is shown in figure
\ref{sqrt2srav}, and looked at the zoom-in of the central part; then at the zoom-in of that zoom-in, and so on, but I could not see
deviation of curve 
$y=\exp_{\sqrt{2},\mathrm u}(x)$ from curve
$y=\exp_{\sqrt{2},\mathrm d}(x)$ .
\vskip 2mm

Searching for the error, as a working hypothesis, I assumed, that the precision of variables 
complex double (which is of order of 15 decimal digits) is not sufficient to see the difference
between
$\exp_{\sqrt{2},d}^{~ 1/2}$ by formula \rf{sqrt2exptet} and 
$\exp_{\sqrt{2},u}^{~ 1/2}$ by formula \rf{Eq2T12}. 
I begun to compute the iterate with a hundred decimal digits;
the exact formulas and abilities of Maple and Mathematica allow this.
The deviation happened to be in 25th digit. This deviation can be characterised with difference
\be
D(x)= \exp_{\sqrt{2},\mathrm u}^{~ 1/2}(x) -  \exp_{\sqrt{2},\mathrm d}^{~ 1/2}(x) 
\eL{sqrt2D}
This difference is shown at the bottom of figure \ref{sqrt2srav}. 
In order to see it, I had to scale it with factor $10^{24}$, as it is marked in the figure.
For
$|z\!-\!3|\!<\!2$, 
function
$D(z)$ 
can be approximated with the 7 parameter fit
\be\!\!
\tilde D(z) \!=\!
2.48\!\cdot\!10^{-25}(z\!-\!2)(4\!-\!z)\Big(
1+0.120(z\!-\!3)+0.006(z\!-\!3)^2\Big) \!\times  \! \nonumber
\\ %\pS 
\!\!\!\!
\sin\!\!\Big(.747\!-\!.068(z\!-\!3)\!+\!0.007(z\!-\!3)^2\!+\! p_{4}\ln(4\!-\!z) \!+\! p_{2}\ln(z\!-\!2) 
\Big) ~ ~ ~ ~\,
\label{cDa}
\label{sqrt2Dfit}
\ee
where
$p_4= 2 \pi / {\ln(2 \ln(2))} \!\approx\! 19.23614904204285$\\% 4712$ \\
and
~ $p_2=\!-2 \pi /\ln^2(2) ~\approx ~ 17.14314817935485$ \\
correspond to periods of the two superfunctions of
$\exp_{\sqrt{2}}$, 
built up on the fixed points 4 and 3, see equations
\rf{q2k4}, \rf{q2P4} 
\rf{sqrt2tetP}. 
Тhis fit provides of order of two significant figures; at figure \ref{sqrt2srav}, the curves for
$D$ and for $\tilde D$ 
almost coincide.
\vskip 2mm

After to plot the bottom curve in figure  \rf{sqrt2srav}, I realised, that it is first case in my life, when the double precision, id est, 15 significant digits, 
happened to be not sufficient to see the difference between two functions, which have no small parameters.
This example gives a good lesson: the numeral calculus with high precision may provide serious evidence to refute, reject (or to verify) a conjecture. 
Whenever the rigorous proof is available of not, the numerical testing should be applied. The TORI axioms provide a good hint for the revision.
\vskip 2mm

It should be noted, that my attempts to use the Maple-10 software for visualisation of difference $D$ by \rf{sqrt2D} failed.
I could not find way to plot graphics with precision better than just ``float'' 
\footnote{The poor precision of the graphic procedures in Maple-10 is described in the poem
\url{https://en.wikisource.org/wiki/Maple_and_Tea}  Maple and tea.
This is one few my texts, that are not yet removed from wikisource with pretext of protection of my author rights.
This is common practice at wikisource and other sites of Wikimedia projects: the Soviet veterans promote sovetism and remove 
texts of anti-Soviet authors with any absurd pretext; often, the claim for the violation of the author rights is used, 
even in cases, when the author gave permission to publish the files providing the free ("copyleft") licence.
The permission to use the author's file is usually removed together the file.
While the soviet veterans act as trolls and vandals in wiki-projects, I loaded the copy also to \url{https://budclub.ru/k/kuznecow_d_j/maple.shtml}
}. 

In order to plot the beautiful figure \rf{sqrt2D}, I had to save values of function
$D$ as a table, and then export this table to the C++ program.
I hope, now there exist more straightforward ways to do the same;
in particular, use of the fit
\ref{sqrt2Dfit} provides the ``quick and dirty'' realisation. 
I expect, using the precise numerical solution, fit
\ref{sqrt2Dfit} can be significantly improved;
the readers are invited to do this as an exercise.
\vskip 2mm

Iterate of a function, regular at some of fixed points, often is singular in another fixed point.
Some exceptions, when the superfunction is expressed with elementary function, are mentioned in Chapter 4.
\vskip 2mm

In such a way, for a given transfer function, there may exist many superfunctions, and some of them may be arguably declared as a ``true'' or ``principal'' superfunction.
For the case of transfer function $T=\exp_{\sqrt{2}}$, the four real-holomorphic superfunctions with various exponential asymptotics are considered in the next section.

~

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\vskip -6pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2sufuplot.png}
\vskip -4pt
\caption{\large 
Four superexponents to base $\sqrt{2}$ \iL{sqrt2sufuplot}}
\vskip 8pt
\end{figure}

\section{Four superexponents to base $\sqrt{2}$}

On the base of consideration of previous chapters, one can built-up the four different real-holomorphic superfunctions for the transfer function $T=\exp_{\sqrt{2}}$.
These superexponentials are shown in figure 
\ref{sqrt2sufuplot} and discussed below. 
\vskip 2mm

Here I compare four functions; in publication \cite{sqrt2}, they are are called $F_{2,1}$, $F_{2,3}$, $F_{4,3}$ and $F_{4,2}$. 
Each of them is real-holomrphic solution of the transfer equation 
\be
F(z\!+\!1)=\Big(\sqrt{2}\Big)^{F(z)}
\ee
The first number in the superscript indicates the limiting value, that the function approaches exponentially; it is any of the two fixed points of the $\exp_{\sqrt{2}}$, id est, either 2 or 4.
The second number in the superscript indicates value of this function at zero.
\vskip 2mm
 
Function $F_{2,1}\!=\!\mathrm{tet}_{\sqrt{2}}$ is tetration to base $\sqrt{2}$; the curve for $F_{2,1}$ is borrowed from the figure \ref{sqrt27a}.
Properties of this function are considered above in this chanter.
It is superfunction of exponent to base $\sqrt{2}$, built up with the regular iteration at fixed point 2. 
As tetration to any other base, it takes value unity at zero.
\vskip 2mm

Function $F_{4,5}\!=\!\mathrm{SuExp}_{\sqrt{2},5}$ refers to formula \rf{Ff45}.
This function, together with its inverse function
 $F_{4,5}^{-1}\!=\!\mathrm{AuExp}_{\sqrt{2},5}$
 is used to built-up iterates of exponent to base $\sqrt{2}$;
 and these iterates grow up infinitely along the real axis.
 Fixed point 4 is used as the asymptotic value at minus infinity.
 With the appropriate translation along the real axis, the condition  $F_{4,5}(0)=5$ is achieved. 
\vskip 2mm

Function $F_{2.3}$ is tetration with displaced argument,
\be
F_{2,3}(z)=\tet_{\sqrt{2}}(z\!+\!z_{2,3})
\ee
\be
z_{2,3}=\mathrm{ate}_{\sqrt{2}}(3\!+\!\mathrm i o)\approx -3.3834692659172254+8.5715740896774228\,\mathrm i
\nonumber
\ee
However, $F_{2,3}$ has the same periodicity, as tetration $F_{2,1}$.
\vskip 2mm

Function $F_{2,3}$ is growing superexponent with displaced argument
\be
F_{4,3}(z)=\mathrm{SuExp}_{\sqrt{2},5}(z\!+\!z_{4,3})
\ee
\be
z_{4,3}=\mathrm{AuExp}_{\sqrt{2},5}(3\!+\!\mathrm i o)\approx 
3.015784890490347\!+\!9.618074521021425 \,\mathrm i
\nonumber
\ee

Along the real axis, functions $F_{2,3}$ and $F_{4,3}$ decrease from 4 at minus infinity to 2 at plus infinity.
In figure \ref{sqrt2sufuplot}, curve $y\!=\!F_{2,3}$ overlaps well with curve $y\!=\!F_{4,3}$. 
The deviation is smaller, than the thickness of lines, and it is small compared to size of atoms, of which this book
(or the screen where it is watched) is built.
In otter to show the deviation, denote it with
\be
d_{42}(z)=F_{4,3}(z)-F_{2,3}(z)
\eL{d42}
Tn figure \ref{sqrt2sufuplot}, the thin line shows  $y\!=\! 10^{24} d_{42}(x)$ ; 
I scale values of this difference for 24 orders of magnitude, to make it visible.
This similarity takes place only in vicinity of the real axis. The functions have different periods, and one of them has singularities; so, they must be pretty different somewhere.
\vskip 2mm

Similarity of functions $F_{4,3}$ and $F_{2,3}$ determines the similarities of corresponding iterates of exponential to base $\sqrt{2}$.
These iterates are 
shown in figures
\ref{iterEq2plot},
\ref{iterEq2map},
\ref{sqrt2eitet} 
and
\ref{sqrt2q2map}. 
For real values of the argument, the half iterates are compared also in figure
\ref{sqrt2srav}.
I expect, for application in physics (where the precision usually does not exceed 20 decimal digits),
any of the two iterates is declared as the ``true iterate''. 
However, for some applications (for example, if the model refers to the complex numbers),
the difference may be important, and the fixed point should be specified.
\vskip 2mm

\section{Complex iterates %экспоненты по основанию $\sqrt{2}$
}
When the draft of the Russian version of this Book had been completed,
I found,
that the book had no maps of the complex iterates. I had declared that I can calculate any real or even complex iterate,
but all the examples refer to real iterate. 
I fill this gap in this section.
Here I describe two iterates number i. As this i appear with Roman font; one may guess, that it is not variable, but a constant, square root of $-1$.
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\vskip -2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2diimap80.jpg}
\vskip -2pt
\caption{\large
$u\!+\!\mathrm i v=\exp_{\sqrt{2},\mathrm d}^{~~~\mathrm i}(x\!+\!\mathrm i y)$
by 
\rf{sqrt2dii} %и \rf{sqrt2uii} 
\iL{sqrt2itemap1}}
\vskip 2pt
\end{figure}
\vskip 2mm

Figures \ref{sqrt2itemap1} and \ref{sqrt2itemap2} 
show the complex maps of iterates
\be\!\!\!\!\!\!\!\!
\exp_{\sqrt{2},\mathrm d}^{~~~\mathrm i}(z)=\mathrm{tet}_{\sqrt{2}}\Big(\mathrm i + \mathrm{ate}_{\sqrt{2}}(z)\Big)
\eL{sqrt2dii}
and
\be\!\!\!\!\!\!\!\!
\exp_{\sqrt{2},\mathrm u}^{~~~\mathrm i}(z)=\mathrm{SuExp}_{\sqrt{2},5}\Big(\mathrm i + \mathrm{AuExp}_{\sqrt{2},5}(z)\Big)
\eL{sqrt2uii}

\begin{figure}[t]
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\end{picture}}
\vskip -2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sqrt2uiimap80.jpg}
\vskip -6pt
\caption{
$u\!+\!\mathrm i v=\exp_{\sqrt{2},\mathrm u}^{~~~\mathrm i}(x\!+\!\mathrm i y)$  по формуле
\rf{sqrt2uii} 
\iL{sqrt2itemap2}}
\vskip 2pt
\end{figure}

Function $\exp_{\sqrt{2},\mathrm d}^{~~\mathrm i}$ by  \rf{sqrt2dii} is built up from the tetration and arctetration to base $\sqrt{2}$, considered in this chapter. 
Function $\exp_{\sqrt{2},\mathrm u}^{~~\mathrm i}$ by \rf{sqrt2uii} is built up from the growing super exponent to base  $\sqrt{2}$ and the corresponding abelexponent;
these functions are considered in chapter 9.
These pairs of functions look similar in vicinity of the interval $(2,4)$, but they are pretty different being evaluated far from this interval.
As one could expect, the $\mathrm i$ th iterates, shown in figures   \ref{sqrt2itemap1} and \ref{sqrt2itemap2}, 
are also similar in vicinity of the interval mentioned, but  far from this interval, they deviate strongly.
\vskip 2mm

As in the case of real iterates, each of considered here complex iterates can be arguably qualified as ``true''. In this sense, the are ``equal''.
In the similar sense, "all animals are equal" in the novel ``Animal's Farm'' by George Orwell
\footnote{
\url{https://msxnet.org/orwell/print/animal_farm.pdf} George Orwell. Animal Farm. 1945. .. {\em ``All animals are equal, but some of them are more equal than others''}.. }
In the movel, soon it happens that ``All animals are equal, but some of them are more equal than others''.
In the similar sense, one of iterates, either   \ref{sqrt2itemap1} or \ref{sqrt2itemap2}, may be ``more equal'', if any
additional criterion arises from some physical reasons, 
specifying, for example, the asymptotic behaviour of the iterate at infinity. If only one of iterates reproduces the required behaviour, this iterate immediately becomes ``more equal'' than another.
\vskip 2mm

The example of this section once again confirms the general observation about non-integer iterates of a function, that have more than one fixed point:
the regular iterate, holomorphic in vicinity of some fixed point, has no meed to be holomorphic at another fixed point. The choice to the fixed point (and choice of the iterate) should involve the additional requirement, that may arise from the applications.
This general rule hold also for complex values of the number of iterate.
\vskip 2mm


I expect, the integer iterates may appear more often that real; and the real iterate may appear more often than complex.
On the other hand, the mathematical formalism should cover an area, which is wider, than that required for the today's applications.
For this reason I consider the case with complex number of iterate as an important example, that shows the abilities of the formalism of superfunctions.
\vskip 6pt

Following the lessons I remember since the Soviet school, I wanted to say that ``the formalism of superfunctions is omni-potnent, because it is true''
\footnote{\url{https://www.marxists.org/archive/lenin/works/1913/mar/x01.htm} V.I.Lenin. The Three Sources and Three Component Parts of Marxism. Lenins Collected Works, Progress Publishers, 1977, Moscow, Vol.19, p.21-28. {.. \em The Marxist doctrine is omnipotent because it is true. ..}
}. 
On the other hand, the First TORI axiom prohibits consideration of omnipotent and almighty concepts in a scientific analysis; such doctrines and concepts are qualified as religious 
\cite{mesto}. 
\vskip 6pt

Various iterates are available for a transfer function with several fixed points.
As soon, as the non-integer iterate of a holomorphic function with several fixed point is required, the additional conditions should be added to the formalism in order to decide, which of the iterates is ``more equal than others''.

\vskip 2mm
~

\section{Not all is done}

In this Book, I describe methods, that can be used to build-up (and evaluate)  superfunctions, and the iterates that can be expressed through the superfunctions and the abelfunctions. The properties of superfunctons and those of the iterates appear as illustrations of the methods. 
I expect the Reader to use these methods not only for the transfer functions, considered in this book.
In particular, this chapter indicates the interesting (from my point of view) branch for the future research, namely, comparison of superfunctions and the corresponding iterates, that are regular in various fixed points, and look for some general criteria: in which cases, the iterates, built up at different real fixed points of some real-holomorphic transfer function, behave in the similar way in the interval between these fixed points. They happened to be very similar in the case of exponent to base $\sqrt{2}$. 
How about other growing transfer functions with two real fixed points?
\vskip 2mm


The Readers are invited to repeat the calculus, described in this Book (and in this Chapter) for other transfer functions.
As an example, I would suggest to experiment with polynomial transfer function.
\vskip 6pt

One example of the polynomial transfer function is shown in raw 5 of table \ref{TableOfSuperfunctions}, $T(z)\!=\!z^b$.
The readers may confirm, that the primary approximation with the regular iteration at the fixed point $z\!=\!1$
gives the series of expansion of the superfunction, that is just expansion of the ``outer'' exponent in superfunction $\exp(b^z)$.
Up to my knowledge, this is the only case, where the primary series by the regular iteration converges. 
The appropriate choice of the 0th approximation should cut the series at the first term.
I expect, many of transfer functions can be treated in such a way, and the corresponding superfunctions and abelfunctions can be built-up.
The goal of this Book not to describe all the examples, but to teach the Readers to to it by themselves. So, it is rather collection of tools, than collection of specific properties of the specific functions. 
\vskip 2mm

Consideration of tetration to base $b\!=\!\sqrt{2}$, presented in this chapter, can be generalised to other values of base; in particular, for real values $1\!<\!b\!<\!\exp(1/\mathrm e)$. 
This generalisation is described in the next chapter.

\chapter{~ Tetration to base $b\!>\!1$}
\baselineskip 20.pt

The previous chapters describe various methods of construction of superfunctions for the transfer functions that have real or complex fixed points.
In particular, the examples of the exponential transfer function are considered, $z\mapsto b^z$ for the cases
$b\!=\!\sqrt{2}\!\approx\! 1.44$,
$b\!=\!\exp(1/\mathrm e) \!\approx\! 1.46$ 
and
$b\!=\!\mathrm e \!\approx\! 2.71~$.

In this chapter, I combine methods, described in the previous chapters, and describe tetration to the real base $b\!>\!1$.
For the real values of base, the interpretation of the superfunction, abelfunction and iterates of the exponent is especially explicit.

\begin{figure}[t]
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\vskip -4pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Tetreal10bx10d.png}
\vskip -1pt
\hskip 24pt
\url{https://en.citizendium.org/wiki/File:Tetreal10bx10d.png}
\vskip -4pt
\caption{ \large
$y=\mathrm{tet}_b(x)$ 
for various
$b$
\iL{tet10bx}}
\end{figure}

\section{Approximation of tetration near zero}

For the base $b$ from the interval
$1\!<\!b \!<\! \exp(1/\mathrm e)$, 
tetration 
$\mathrm{tet}_b$ can be evaluated with regular iteration at the lowest (smallest) positive fixed point of the exponent to this base. 
For
$b\!=\!\exp(1/\mathrm e)$, it can be evaluated with the exotic iteration by equation \ref{e1etet}. 
For
$b\!>\!1/\exp(1/\mathrm e)$,
the representation through the Cauchy integral can be used. In such a way, all the domain
$b\!>\!1$ 
is covered with the efficient algorithms for evaluation of 
$\mathrm{tet}_b$. 
\vskip 2mm

For real values of the argument, graphic $y\!=\!\mathrm{tet}_b(x)$ is shown in figure \ref{tet10bx}
versus $x$ for various values of base $b\!>\!1$. 
Similar plot for arctetration is shown in figure \ref{ate10bx}.

\vskip 2mm
Figure \ref{tet10bx} for tetration and figure \ref{ate10bx} for arctetration
are generated, using the special approximation of tetration for 
$b\!<\!3$, 
by function $\mathrm{fit1}$, 
defined with
\vskip 6pt
\baselineskip 17.4pt

$d\!=\!\ln(b)$
\vskip 6pt

$q\!=\!\sqrt{d}$
\vskip 6pt

$c_0\!=\! -1.0018+
\frac{0.1512848482(1.+33.0471529885 q-3.51771875598 d) q}
{1+3.2255053261256337 q}
+   \frac{\ln(2)-\frac{1}{2}}{d}$
\vskip 6pt

$c_1\!=\!
1.1 - 2.608785958462561(1 - 0.6663562294911147q) q - \frac{\ln(2)-\frac{5}{8}}{d}$
\vskip 6pt

$c_2\!=\!
-0.96 + 3.0912038297987596 \frac{(1\!+\!0.60213980487853d)q}{1 \!+\! 4.24046755648d}  + 
 \frac{\ln(2)-\frac{2}{3}}{d}$
\vskip 6pt

$c_3\!=\! 1.2\! -\! 10.44604984418533 \frac{(1\!+\!0.213756892843 q\!+\!0.369327525447 d) q}{1+4.9571563666 q + 7.702332166 d}
- \frac{ \ln(2)- \frac{131}{192}}{d} $

\be
\mathrm{fit1}_b(z)\!=\! (1\!+\!c_0z\!+\!c_1z^2\!+\!c_2z^3\!+\!c_3z^4)(z\!+\!1) 
+ \ln(z\!+\!2)  
-\frac{\ln(2)}{d} (1\!+\!z) \nonumber \\
\label{fit1}
\ee

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\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Ater01.png}
\vskip -8pt
\caption{%\large
$y\!=\!\mathrm{ate}_b(x)$ 
for various
$b$ \iL{ate10bx}}
\vskip -8pt
\end{figure}

In order to get approximation \rf{fit1}, I expanded the expression
\be
\frac{\mathrm{tet}_b(z) - \ln(z\!+\!2) + \ln(2)/\ln(b) }{1\!+\!z}
\ee
into the Taylor series with powers of  $z$ for various values of (new) parameter  $d=\ln(b)$. 
Coefficients of this expansion are approximated as functions of parameter  $d$. 
Then, tetration is expressed through this expansion.
The series is truncated; only few terms are taken into account. 
For 
$|z|\le1/2$, the resulting approximation provides few correct decimal digits of $\mathrm{tet}_b(z)$.
This approximation is used for $|\Re(z)|\le 1/2$; for other values of $z$, 
value of tetration is represented through its values at the appropriate argument, using either
\be
\mathrm{tet}_b(z)=b^{\mathrm{tet}_b (z-1)}
\eL{tetbminus}
or
\be
\mathrm{tet}_b(z)=\log_b\! \big(\mathrm{tet}_b(z\!+\!1)\big)
\eL{tetbplus}
dependently on  signum of 
$\Re(z)$. 
In such a way, tetration can be evaluated with few decimal digits for moderate values of 
$|\Im(z)|$ 
and
$|\Im(b)|$.
Yet, I have no similar approximation for larger values of  $|\Im(b)|$, 
although, the Cauchy integral can be generalised for various, even complex values of $b$;  
the example is considered below in chapter 18.
The range of validity of approximation $\mathrm{fit1_b}$ is wide; In particular, at
$b\!<\!5$, 
the approximation provides of order of four significant figures.
This is sufficient precision for plotting of the camera-ready pictures, the defects of this approximation are not seen 
in figure 
\ref{tet10bx}. However, at $b\!=\!10$, the approximation is a little bit worse; for this value, the primary representation through the Cauchy integral is used. This refers to only single curve in the picture. 
\vskip 2mm

Figure \ref{tet10bx} shows behaviour of tetration of real argument at various values of base $b\!>\!1$:
\vskip 2mm

In the interval $-2\!\le\! x\! \le\! -1$, tetration $y\!=\! \mathrm{tet}_b(x)$ has negative values and grows while  $b$ increases.
\vskip 6pt

In the interval $-1\!\le\! x\! \le\! 0$, tetration $y\!=\! \mathrm{tet}_b(x)$ has positive value and grows with increase of $b$.
\vskip 6pt

At $x\!>\!0$, tetration $y\!=\! \mathrm{tet}_b(x)$ has positive values and grows with increase of $b$.
\vskip 6pt

In the limiting case $b \! \rightarrow \!1$, the curve  $y\!=\! \mathrm{tet}_b(x)$  approaches the asymptotics $x\!=\!-1$ and $y\!=\!1$.
\vskip 6pt

In the limiting case $b \! \rightarrow \! \infty$, curve $y\!=\! \mathrm{tet}_b(x)$  approaches asymptotics $x\!=\!-2$ and $x\!=\!0$, 
and also $-2\!<\! x \!<\! 0$  at the abscissa axis.
\vskip 6pt

At all $b \! > \!1$, tetration $\mathrm{tet}_b(x)$ is monotonic function, and curve  $y\!=\! \mathrm{tet}_b(x)$ 
passes though points $(-1,0)$, $(0,1)$ and $(1,b)$.
\vskip 6pt

At $b \! = \! \exp(1/\mathrm e)$, the line $y\!=\!\mathrm{filog}(1/\mathrm e)\!=\!\mathrm e$ becomes asymptotics. 
this line is added to the rectangular grid at integer values of abscissas and ordinates.
\vskip 2mm

At $1\! <\! b \! \le \! \exp(1/\mathrm e)$, with grow of $x$ the curve $y\!=\! \mathrm{tet}_b(x)$ approaches the horizontal 
asymptotics  $y=\mathrm{filog}\big(\ln(b)\big)$.
\vskip 2mm

Function filog expresses fixed points of logarithm as function of logarithm of its base. This function is considered in the next chapter. It is essential for evaluation of tetration of complex base.
While, suggest to compare the results of the previous chapters for the real base. This is matter of the next section.

\JP{

The inverse function of tetration, id est, arctetration, can be approximated in a way, similar to that of tetration.
I suggest, that the Readers prepare (and test) such an approximation by themselves; I did not do it.
Instead, for plotting of figure \ref{ate10bx}, I just reflected the curve of \ref{tet10bx} from the line $y\!=\!x$.
\vskip 2mm

For $x\!>\!-2$, the following relation takes place:
\be 
\mathrm{ate}_b \Big( \mathrm{tet}_b(x) \Big)=x
\ee
In addition, for $b\!>\!\exp(1/\mathrm e)$ and at least for real $x$ the following relation is valid:
\be 
\mathrm{tet}_b\Big( \mathrm{ate}_b(x) \Big) =x
\ee
The same relation is valid also for
$~1\!< \!b\! \le \! \exp(1/\mathrm e)~$, $~x \! < \! \mathrm{filog}\big(\ln(b)\big)~$.
\vskip 4mm

In wide range of base $b$ and argument $z$ iterates of exponent can be expressed through tetration and arctetration,
\be
\exp_b^{~n}(z)= \mathrm{tet}_b\Big(n+\mathrm{ate_b}(z) \Big)
\ee 
As in other cases, the number $n$ of iterate has no need to be integer, but $z$ should belong to the range of holomorphizm of 
function $\mathrm{ate}_b$. Then, at $b\!>\!\exp(1/\mathrm e)$, value $n\!+\!\mathrm{ate}_b(z)$ belongs to the range of holomorphism of tetration $\mathrm{tet}_b$.
}%JP

\section{Various bases of exponent and the iterates}

In this section, the iterates of exponent are compared. These iterates are considered in the previous sections.
Below, I overview the results.
\vskip 2mm

Iterates of exponent to bases $b\!=\!\mathrm e$, $b\!=\!\exp(1/\mathrm e)$ and $b\!=\!\sqrt{2}$ are shown in figure 
\ref{e1e14} 
as functions of the real argument for various values of number $n$ of iterate. The curves are drawn for
$n\!=\!$ $-2$, $-1$, $-0.9$, $-0.5$, $-0.1$, $0$, $0.1$, $0.5$, $0.9$, $1$, $2$.  
\vskip 2mm
 
\begin{figure}
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\vskip -5pt
\hskip 26pt
\url{https://mizugadro.mydns.jp/t/index.php/File:E1e14z600.jpg}
\vskip -12pt
\caption{
\large 
$y\!=\!\exp_b^n(x)$ 
for various $n$ at
$b\!=\!\mathrm e$, 
$b\!=\!\exp(1/\mathrm e)$,
$b\!=\!\sqrt{2}$
\iL{e1e14}}
\end{figure}

The upper picture in figure \ref{e1e14} represents the case $b\!=\!\mathrm e$; iterates of the natural exponents are shown.
These iterates are calculated through the natural tetration tet and natural arctetration ate, I repeat this formula once again:
\be
y=\exp^n(x)=\mathrm{tet}\Big( n + \mathrm{ate}(x) \Big)
\ee
Similar pictures can be plotted also for other values of base
$b\!>\exp(1/\mathrm e)$. 
For these values of base, the real iterates are real-holomorphic functions at least in some vicinity of the real axis. In this area,
the iterates of the exponent are so smooth, as the exponent itself.
\vskip 2mm

As the base $b$, the width of the strip along the real axis (where the iterates are holomorphic) decrease;
at base 
$b=\exp(1/\mathrm e)$ 
all the curves for various iterates pass through the fixed point  $\mathrm e$. 
This case is shown in the central picture of figure \ref{e1e14}. 
Then, the non-integer iterates of exponents at argument, larger than $\mathrm e$, 
are not anymore holomorphic extensions of those for argument, smaller than $\mathrm e$. 
In order to stress this, the curves, plotted through tetration and arctetration, are shown with dashed lines
\be
y=\mathrm{exp}_{b,\mathrm d}^{~ n}(x)=\mathrm{tet}_b\Big(n+\mathrm{ate}_b(x)\Big) 
\ee
while the solid lines correspond to the iterates, expressed through the growing superexpponent SuExp,
\be
y=\mathrm{exp}_{b,\mathrm u}^{~ n}(x)=\mathrm{SuExp}_{b,3}\Big(n+\mathrm{AuExp}_{b,3}(x)\Big) 
\ee
Tetration $\mathrm{tet}_b$ to base $b\!=\! \eta\!=\!\exp(1/\mathrm e)$ is determined with
\rf{e1etet},
and the arctetration to this base can be evaluated through
\rf{e1eG1}.
In the similar way, the growing superexpenent is determined 
by \rf{e1eSuExp}
with
$\mathrm{SuExp}_{b,5}\!=\!F_3$, 
and the corresponding abelexponent can be evaluated with
\rf{e1eAuExp}
at
$\mathrm{AuExp}_{b,3}\!=\!G_3$.
\vskip 2mm

For the real base $b$, at $1\!<\!b\!<\!\exp(1/\mathrm e)$, 
The exponent has two real fixed points
(See figures \ref{ExpQ2plot}, \ref{expe1eplot}). 
Each to these fixed points can be used for the regular iteration.
However, the non-integer iterate, regular at one fixed point, is singular at another one.
At the bottom picture of figure  \ref{e1e14}, value  $b\!=\!\sqrt{2}$ is chosen. 

Dashed lines corresponds to\be
y=\mathrm{exp}_{b,\mathrm d}^{~ n}(x)=\mathrm{tet}_b\Big(n+\mathrm{ate}_b(x)\Big) 
\ee
while the solid curves refer to
\be
y=\mathrm{exp}_{b,\mathrm u}^{~ n}(x)=\mathrm{SuExp}_{b,5}\Big(n+\mathrm{AuExp}_{b,5}(x)\Big) 
\ee
For non-integer number $n$ of iterate, the fixed point 2 or 4 limits the range of holomorphism of each iterate.
In the interval from 2 to 4, each of the two iterates is holomorphic, and the difference between these iterate is very small, of order of $10^{-24}$,
see figure  \ref{sqrt2srav}. Due to this smallness, the dashed lines at the bottom picture of figure \ref{e1e14} seem to coincide with the corresponding solid lines.
\vskip 2mm

Similar illusion takes place for the central part of figure \ref{e1e14}; 
the dashed lines seem to be continuations of the corresponding solid lines.
The example with exponent show, that, in order to specify the non-integer iterate,
one should choose, establish the asymptotic behaviour of the superfunction in the complex plane.
Over-vice, there may exist various solutions, and each of them arguably can be declared as the ``true'' one.
In order to specify the superfunctions, they are considered for the complex argument.

\JP{
\vskip 2mm

Algorithms of evaluation of functions, described in this Book, are implemented as complex double.
The URLs to get the algorithm are specified under each complex map or explicit plot of the Book.
For complex argument, the small difference ($\sim10^{-24}$) mentioned above, become visible even with complex double precision. 
However, the same algorithms can be used also for evaluation of suerfunctions, abelfunctions and iterates at real values of the argument.
}
\section{%Непрерывность 
Dependence of tetration on its base
}

Graphics of tetration of real argument, shown in figure  \rf{tet10bx}, allow to guess, that the dependence of tetration on the base (at fixed argument)
is continuous (and, perhaps, even holomorphic) function.
In order to show, that this refers not only to the real values of the argument, figure \ref{e1e09} shows 
the complex maps of tetration 
for $b\!=\!1.5$ at left,
for $b\!=\!\exp(1/\mathrm e)\! \approx\! 1.44$ at center, and
for  $b\!=\!\sqrt{2}\!\approx\! 1.41$ at right.

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\vskip -2pt
\hskip 20pt
\url{https://mizugadro.mydns.jp/t/index.php/File:E1efig09abc1a150.png}
\vskip -4pt
\caption{%\large
$u\!+\!\mathrm i v\!=\!\mathrm{tet}_b(x\!+\!\mathrm i y)$ for $b\!=\!1.5$, $b\!=\!\exp(1/\mathrm e)$ and $b\!=\!\sqrt{2}$ \iL{e1e09}}
\end{figure}
\vskip 2mm

All the 3 maps in figure \ref{e1e09} look similar, although different algorithms are used for evaluation of tetration.
 %at the specified values of base. 
In principle, tetration to base $b\!=\!\exp(1/\mathrm e)$, for moderate values of the imaginary part of the argument, 
could be evaluated also through tetration with a little bit smaller or a little bit larger values of the base $b$,
as limit $b \rightarrow \exp(1/\mathrm e)$,
using the corresponding representation through the integral Cauchy (for bigger values) or with regular iteration (for smaller values).
The Readers are invited to calculate tetration to base $b\!\approx\! \exp(1/\mathrm e)$ and estimate, 
how many significant figures can one achieved in such a way, assuming, that the arithmetics with finite precision is used. % (for example, complex double).
\vskip 2mm
\RU{
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\vskip -12pt
\sx{.3}{\begin{picture}(200,810)
\put(0,0){\ing{nepreryvnost}}
\end{picture}}
\vskip -8pt
\caption{Неразрывность \label{nepreryvnost}}
\vskip -8pt
\end{wrapfigure} }%RU

At $b \!\rightarrow\! \exp(1/\mathrm e)$, the efficiency of evaluation of tetration (both through the Cauchy integral and through the regular iteration) reduces. For this reason, Henryk Trappmann had expected, that $\mathrm{tet}_b$ is not continuous function of $b$
at point  $b\!=\!\eta\!=\!\exp(1/\mathrm e)$.
For this reason, Henryk wanted the asymptotic expansion namely for $\eta\!=\!\exp(1/\mathrm e)$, and it had been done
\footnote{
That happened soon after the article about four real-holomorphic superexponentials to base $\sqrt{2}$ had been submitted to 
Mathematics of computation \cite{sqrt2}; as usually, the appetite comes while eating.}.
I had suggested the expansion \rf{e1e24} and plotted pics for $b\!=\!\exp(1/\mathrm e)$, and Henryk had arranged a lot of mathematical deduction around it \cite{e1e}.
\vskip 6pt

I hope, the Reader already understands, how to guess the expansion for the exotic iterates, and can write the similar  expansion of superfunction
for any other transfer function, as soon as such an superfunction will be requested for any application.
The inversion of the series gives the expansions (and the precise approximations) for the corresponding abelfunctions.
\vskip 6pt

I hope, with the tools above, the colleague can evaluate any iterate of any holomorphic transfer function, not only real, but also complex.
The examples with iterates number $\mathrm i$ are shown in figures 
\ref{sqrt2itemap1} and
\ref{sqrt2itemap2} for the transfer function $T=\exp_{\sqrt{2}}$.
\JP{
\footnote{
Here, $i$ is variable, that can take various values; often, this letter is used as integer counter.
However $\mathrm i\!=\!\sqrt{-1}$ is, contrary, constant, that cannot take any other value.}
}
\vskip 6pt

After publications of the results presented above, the colleagues at the Henryk's forum had agreed, that the complex iterates, and in particular, those of
the exponent to various real base $b\!>\!1$, can be evaluated in a pretty regular way. However,
There were some doubts about iterates of the exponent to the complex base.
This case is considered in the next chapter.
\chapter{~ Tetration to complex base}
\baselineskip 19pt

\JP{
\begin{wrapfigure}[18]{r}{100px}
\sx{.4}{\begin{picture}(300,450)
\put(0,0){\ing{tetco}}
\end{picture}}
\caption{ Комплексное основание \label{tetco}}
\end{wrapfigure}
}%

Here, tetration to complex base is considered. Id est, superfunction for the transfer function $T(z)=b^z$, where base $b$ is not 
real. In order to handle this case, I need an additional special function "filog"; it is also described in this chapter.
\vskip 6pt

In principle, the superfunction of the exponent to complex base can be constructed with regular iteration, in the similar way, as the tetration to base
$\sqrt{2}$ is constructed in Chapter 16.
However, the important question is, at which of the fixed points, the superfunction should be regular, and which of possible superfunction should be qualified as tetration. 
\vskip 6pt

In addition, at some values of the base, the exponential asymptotic solution has real part of the increment zero or close to zero; this makes the application of regular iteration difficult, if at al.
One of such cases is considered below in more details, as an example.
The representation of tetration through the Cauchi integral is not sensitive to the real part of the asymptotic increment of the solution; so, such a representation 
gives the efficient way of evaluation of tetration to complex base. However, the asymptotic behaviour of tetration in the upper and in the lower complex plane should be specified.

\vskip 6pt

The main idea of this chapter is to make superfunction that approaches one of the fixed points of the exponent at the upper half of the complex plane, and to another fixed point at the lower part of the complex half plane, using the assumption, that, for the complex base, the imaginary part of the asymptotic increment is not zero. 
\vskip 6pt

Question about the fixed points is important (as in the case of any other superfunction), and it should be considered.
This consideration is presented in the following section.


\section{Fixed points of logarithm}
\baselineskip 18pt
This section is dedicated to relation between the base $b$ of the exponential and its fixed point $L$.
Let $L=\mathrm{filog}(B)$ be solution of equation
\be
\ln(L)=BL
\eL{LL}
Complex map of function filog is shown in figure
\ref{filogmap}.
The zoom-in of the central part of this map is shown in figure 
\ref{filogzo2}.
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\vskip -4pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Filogbigmap100.png}
\vskip -8pt
\caption{\large $u\!+\!\mathrm i v\!=\!\mathrm{filog}(x\!+\!\mathrm i y)$ ,
as solution
$L\!=\!\mathrm{filog}(B)$
of 
\rf{LL} \label{filogmap} } 
\vskip 8pt
\end{figure}
Let
\be
B=\ln(b)
\eL{Bb}
Then  $\mathrm{filog}(B)$ determines the fixed point $L_1$ of logarithm to base $b$; another fixed point $L_2$
is determined with the complex conjugation:
\be
L_1&=&\mathrm{fllog}(B)  ~ ~\iL{L1filog}\\
L_2&=&\mathrm{filog}(B^*)^*
\eL{L2filog}

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\vskip 16pt

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\put(816,524){\sx{2.8}{\rot{56}$u\!=\!1.6$ \ero} }
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\put(94,99){\sx{3}{\rot{41}$v\!=\!-2$ \ero} }
\put(162,50){\sx{3}{\rot{57}$v\!=\!-1$ \ero} }
\put(242,30){\sx{3}{\rot{75}$v\!=\!0$ \ero} }
\put(315,30){\sx{3}{\rot{87}$v\!=\!1$ \ero} }
\end{picture}}
\vskip -2pt
\hskip 18pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Figlogzo2t.jpg}
\vskip -6pt
\caption{%Zoomin from figure \ref{filogmap} \iL{filogzo2} 
$u\!+\!\mathrm i v=\mathrm{filog}(x\!+\!\mathrm i y)$, 
zoom-in from the central part of figure
\ref{filogmap} \iL{filogzo2} 
}
\vskip -406pt
\end{figure}

\rule{0pt}{200pt}

\newpage

Function filog can be expressed through the special function Tania by \rf{taniaprim} as follows:
\be
\displaystyle \mathrm{filog}(z)= \frac{\mathrm{Tania}\big(\ln(z)-1- \pi \mathrm{i}\big)}{-z} 
\ee
Note, that here, namely Tania is used, but not WrightOmega, which looks similar to Tania in vicinity of the real axis.
The Readers are invited to try to reproduce figure  \ref{filogmap}, using WrightOmega instead of Tania, and look,
what does it give instead of the beautiful map.
\vskip 2mm

Function filog determines the fixed points of logarithm (which are also fixed point of the exponent) to base $b$:
\be
L_1&=&\mathrm{filog}\big(\ln(b)\big)  ~ ~\iL{L1filob}\\
L_2&=&\mathrm{filog}\big(\ln(b^*)\big)^*
\eL{L2filob}
At $\pm \mathrm i \infty$, 
tetration should approach these fixed points.
This assumption 
to use the Cauchy  integral for the definition (and evaluation)
\section{Tetration to the Sheldon base}
\baselineskip 17pt

This section describes the tetration to the Sheldon base,
\be
b=1.52598338517+0.0178411853321 \,\mathrm i
\eL{sheldonS}
This number is named after Sheldon Levenstein. In 2015, Sheldon had expected, that the namely this base causes difficulties at evaluation of tetration.
It was the only request from colleagues to evaluate tetration to the specific complex base;
and this request had been fulfilled. 
\vskip 6pt

Complex map of tetration to the Sheldon base is shown in the top picture of figure 
\ref{sheldonmap}. Explicit plot of this function is shown in 
the central picture of that figure.
The bottom picture shows the explicit plot of tetation to Sheldon base along the imaginary part. This tetration has complex values; so, the graphics are drawn for the real and for the imaginary parts.
\vskip 6pt

In this section I assume, that value of $b$ is determined with equation \rf{sheldonS}. I consider this as an example; 
tetration to other values can be calculated 
in the similar way.
\vskip 6pt

It is convenient to define $B=\ln(b)$. Then, the fixed points of logarithm to base $b$, id est, solutions  $L$ of equation  $\ln_b(L)\!=\!L$, can be expressed through function filog, described in the previous section:
\be
\!\!\!\!\!\!\!\!\! L_1 = \mathrm{filog}(B) ~ \approx 2.0565398441043761 \!+\! 1.1445267140098765 ~ \mathrm i \label{sheldonL1}\\
\!\!\!\!\!\!\!\!\! L_2\!=\! \mathrm{filog}(B^*)^*\! \approx 2.2284359658711805 \!-\! 1.3507994961102865 ~ \mathrm i
\label{sheldonL2}
\ee
\newpage
\begin{figure}[h]
\hskip 9pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Tetsheldonmap03.png}
\vskip 4pt
\sx{.7}{\begin{picture}(602,211)
\put(10,10){\ing{tetsheldonmap}}
\put(2,206){$y$}
\put(2,188){$8$}
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\put(2,128){$2$}
\put(2,108){$0$}
\put(-6,88){$-2$}
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\put(-6,48){$-6$}
\put(-6,28){$-8$}
\put(-1,0){$-30$}
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\put(507,0){$20$}
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\put(607,1){$x$}
\multiput(24,180)(118,-1){5}{$v\!=\!1.2$}
\multiput(70,180)(118,-1){5}{$u\!=\!2$}
\multiput(112,158)(118,-1){5}{\rot{-50}$v\!=\!1$\ero}
\multiput(102,142)(118,-1){5}{\rot{-35}$v\!=\!0.8$\ero}
\put(336,107){\rot{-11}$v\!=\!0$\ero}
\multiput(206, 92)(119,-22){4}{$v\!=\!-1$}
\multiput(70, 84)(119,-22){4}{$u\!=\!1.2$}
\multiput(110, 76)(119,-22){4}{$v\!=\!-1.4$}
\put(420,199){\sx{1.3}{$u+\mathrm i v \approx 2.0565+1.1445 \,\mathrm i$}}
\put(30,26){\sx{1.3}{$u+\mathrm i v \approx 2.2284-1.3508 \,\mathrm i$}}
\end{picture}}
\vskip 12pt
\JP{
\caption{
 $u\!+\!\mathrm i v\!=\! \mathrm{tet}_b(x\!+\!\mathrm i y)$ 
for
$b\!\approx\!
   1.52598%338517 
\!+\! 0.01784%11853321
\,\mathrm i $, 
see \rf{sheldonS} \label{sheldonmap} }
\end{figure}

\begin{figure}[h]
}%JP

\url{https://mizugadro.mydns.jp/t/index.php/File:Shelr80.png}
\vskip 6pt
\sx{.162}{
\begin{picture}(3202,1304)
\put(0,0){\ing{tetsheldor4}}
\put(168,1286){\sx{6.7}{$y$}}
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\put(330,800){\sx{7.1}{\rot{20}$y\!=\! \Re\big(\mathrm{tet}_b(x)\big)$\ero}}
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\put(320,636){\sx{7.1}{\rot{6}$y\!=\! \Im\big(\mathrm{tet}_b(x)\big)$\ero}}
\end{picture}}
\vskip 12pt
\JP{
\caption{ $y\!=\!\Im(\mathrm{tet}_b(x))$ and
$y\!=\!\Re(\mathrm{tet}_b(x))$ for $b\!\approx\!
   1.52598%338517 
\!+\! 0.01784%11853321
\,\mathrm i $, thick curves; the thin curve reproduces graphic 
$y\!=\!\mathrm{tet}_{1.5}(x)~$ from figure \ref{tet10bx}
\iL{tetsheldor} }
\vskip 8pt
\end{figure}
\vskip 10pt

\begin{figure}[h]
}%JP

\url{https://mizugadro.mydns.jp/t/index.php/File:Shelima600.png}
\vskip 4pt
\sx{2.1}{\begin{picture}(602,62)
\put(0,0){\ing{tetsheldoi}}
\put(97,59){\sx{.6}{$y$}}
\put(97,49){\sx{.6}{$2$}}
\put(97,39){\sx{.6}{$1$}}
\put(97,29){\sx{.6}{$0$}}
\put(92,19){\sx{.6}{$-1$}}
\put(92, 9){\sx{.6}{$-2$}}

\put(15,25){\sx{.6}{$-8$}}
\put(35,25){\sx{.6}{$-6$}}
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\put(75,25){\sx{.6}{$-2$}}
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\put(160,25){\sx{.6}{$6$}}
\put(180,25){\sx{.6}{$8$}}
\put(199,25){\sx{.6}{$x$}}

\put(10,56){\sx{.7}{$y\!=\! \Re\big(\mathrm{tet}_b(\mathrm i x)\big)$}}
\put(10,11){\sx{.7}{$y\!=\! \Im\big(\mathrm{tet}_b(\mathrm i x)\big)$}}
\end{picture}}

\vskip 4pt
\caption{
Tetration to base $b \!=\! 1.52598338517 \!+\! 0.0178411853321
\,\mathrm i ~$: $  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~$
Top map: 
$u\!+\!\mathrm i v\!=\! \mathrm{tet}_b(x\!+\!\mathrm i y)$ ~;~ ~
$~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~ ~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~~~~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~ ~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ $
Intermediate plot: 
$y\!=\!\Im(\mathrm{tet}_b(x))$ and
$y\!=\!\Re(\mathrm{tet}_b(x))$, thick curves; $~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~~ ~ ~ ~ ~ ~ ~ ~ ~$
the thin curve shows 
$y\!=\!\mathrm{tet}_{1.5}(x)\,$ from figure \ref{tet10bx} ;$~ ~ ~ ~ ~~~ ~~  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~$
Bottom plot: 
$y\!=\!\Im(\mathrm{tet}_b(\mathrm i x))$ and
$y\!=\!\Re(\mathrm{tet}_b(\mathrm i x))$  $~ ~ ~$
 \iL{sheldonmap}
 \label{tetsheldor}
 \label{tetsheldoi}
  }
 \vskip -190pt
\end{figure}

~
\vskip 80pt
~
\newpage

where constant $B$ is evaluated as follows:
\be
B=\ln(b)\approx 0.4227073870410604 + 0.0116910660021443 \mathrm i
\ee

Solution $F$ of the transfer equation $b^{F(z)}=F(z\!+\!1)$ with asymptotics
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F(z)\!&\!=\!&\!L_1\!+\! \exp( k_1 z \!+\! \phi_1) + \mathcal O(\exp( 2 k_1 z))~ ~ \text{at} ~ ~ \Im(z)\!\rightarrow  \infty ~
\label{sheldonFasUp}
\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F(z)\!&\!=\!&\!L_2\!+\! \exp( k_2 z \!+\! \phi_2) + \mathcal O(\exp( 2 k_2 z))~ ~ \text{at} ~ ~ \Im(z)\!\rightarrow\! - \infty ~
\label{sheldonFasDown}
\ee
can be considered in the similar way, as for tetration to real base, larger than the Henryk base
$\eta\!=\!\exp(1/\mathrm e)$. 
Even the same contour of integration can be used.
Subslitution of the asymptotic solutions \rf{sheldonFasUp}, \rf{sheldonFasDown} into the transfer equation determines the increments
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
k_1 \!&\!=\!&\! \ln(L_1 b) \approx \! -0.0047589243931785 \!+\! 0.5354935770338939\, \mathrm i 
\label{sheldonkup}\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
k_2 \!&\!=\!&\! \ln(L_2 b) \approx ~  ~ 0.0970758595007548 \!-\! 0.517289596155984\,\mathrm i
\label{sheldonkdown}\ee
The solution has quasiperiod
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
P_1= \frac{2 \pi \mathrm i} {k_1} \approx 
11.7325200133916496
\!-\! 
0.1042667514229599 \,\mathrm i 
\label{sheldonPup}
\ee
in the upper part of the complex plane, and quasiperiod
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
P_2=\frac{2 \pi \mathrm i}{k_2} \approx  
11.7331504449085493
 \!-\! 
2.2018723603861230 \, \mathrm i
\label{sheldonPdown}
\ee
in the lower part of the complex plane,
\vskip 2mm

The properties above are sufficient to express the solution $F(z)$ of the transfer equation
\be
F(z\!+\!1)=\exp\Big(B\, F(z)\Big)
\eL{sheldonTra}
through the Cauchy integral.
This construction is quite analogous to that for the tetration to base
$\mathrm e$,
described in chapter 14; therefore I do not repeat here the description of the contour of integration nor the iterational procedure,
that provides the approximations of the solution.
The tetration is expressed through the solution $F$ in the following form:
\be
\mathrm{tet}_b(z)=F(z_1+z)
\eL{sheldonTetDef}
where $z_1$ is solution of equation  $F(z_1)=1$.
Using equation 
\rf{sheldonTra}, 
the solution can be extended at least to the right hand side of the complex plane.
As for the left hand side, in the Second quadrant of the complex plane, the branch points and the cutlines appear.
These cutlines appear at the use of the logarithmic function to extend the solution to the direction of negative values of the real part of the argument. 
\vskip 6pt

Unfortunately, the most of cut lines do not fit the frame of map in figure  \ref{sheldonmap}, althogh I make it in the whole width of the page.
The asymptotic  \rf{sheldonFasUp} indicates, that these cuts are unavoidable, while the imaginary part of quasi period $P_1$ is negative.
Each time, when the tetration takes value zero, there is branch point at value of the argument, for unity smaller.
These cuts are unavoidable also for other tetrations, while formula
\be
F(z)={\log_b}^m (F(z\!+\!m))
\ee
is used for some integer $m$ such that value $z\!+\!m$ belongs to the strip $\{z \in \mathbb C : |\Re(z)|\le 1/2 \}$.
In this sense, the tetration to the Sheldon base is similar to tetration to the real base.

\section{Shell-Thron region}
\baselineskip 19pt

For moderate values of argument  $z$, tetration  $\mathrm{tet}_b(z)$ looks as a smooth function of base $b$. 
At large values of $\Re(z)$, dependently on the base, tetration either has complicated, quasi-chaotic behaviour, 
or approaches some of the fixed point of the corresponding logarithm.
At the site ``Eretrande'',
the range of values of base $b$, for which the tetration approaches its limiting value, is called Shell-Thron region
\footnote{ Shell-Thron region:\\ \url{https://math.eretrandre.org/hyperops_wiki/index.php?title=Shell-Thron_region}. }.
Such a name seems to be commonly accepted, and I even suggest the Russian literal translation
``Область Тронной Ракушки''
for the Russian version of this Book.
However, yet, it is difficult to predict, how convenient and stable are these names.
\vskip 2mm

In this Book, I am interested mainly the tools for evaluation of superfunctions, and resolving the paradoxes, that 
are discussed among colleagues.
Consideration of many examples, that cause no doubts, fall out of scope of this Book.
\vskip 2mm

In principle, at the iterates of exponent to complex base, we have to deal with the 6-dimensional space.
Coordinates of this space are the real and imaginary parts of base $b$ (or real and imaginary parts of its logarithm,
$B\!=\!\ln(b)$), 
the real and imaginary parts of the argument,and
the real and imaginary parts of the number of iterate.
There may be many interesting effects hidden in the 6-dimensional space.
The detailed description of these effects may require a special book, dedicated namely to tetration to complex base.
Since year 2010, Henryk Trappmann tries to compose such a book \cite{bitbucket} 
(I had promised to provide all the algorithms and pictures he needs for this, and all the pictures requested up to date are already plotted an included there). I hope, one day Henryk gets good grant for this activity and will be able to polish the text to the state he considers as satisfactory.
While I still see no principal problem, that cause serious difficulty at the evaluation of the corresponding iterates; so, I do not include parts of 
\cite{bitbucket} to this Book. For this reason, analysis of the Shell-Thron region is presented here in a declarative form.
\vskip 2mm

I hope, the Readers can plot all the pictures that are necessary for illustration of tetration to the complex base,
using the tools described above. However, if any difficulties of paradoxes appear in the analysis, I shall try to consider and to resolve them.
This point of view is described by the Russian writers Arkadi \& Boris Strugatski in the novel ``Monday begins on saturday''
\cite{ponedelnik}: {\em .. It's nonsense to look for a solution if it already exists. We are talking about how to deal with a problem that has no solution. ..}
\vskip 2mm

I hope, the Readers can evaluate tetration to other bases by themselves, using the tools any examples above. Following the idea mentioned, I continue to deal with cases that are believed to have ``no solution''.
One of the such cases refer to the superfunction of tetration; let it be called ``pentation''.
In order to bring is to the textbook case (see figure \ref{textbookcase}), this function, among other ackermanns, is considered in the next chapter.

\begin{figure}[b]
\hskip 140pt
\sx{.4}{\begin{picture}(200,200)
\put(0,0){\ing{textbookcase}}
\end{picture}}
\caption{\large Textbook case \label{textbookcase}}
\end{figure}


\chapter{~ Ackermanns \label{Ackermanns}}
%\baselineskip 17pt
\baselineskip 16pt

\begin{wrapfigure}{r}{118pt}
\sx{.4}{
\begin{picture}(100,100)
\put(0,300){\ing{Ackermann_Wilhelm}}
\put(20,230){\sx{2.2}{W.Ackermann}}
%\put(280,90){\rot{90} \sx{2}{test} \ero}
\put(280,100){\rot{90}\sx{1.25}{\url{https://mizugadro.mydns.jp/t/index.php/File:Ackermann.jpg}}\ero}
\end{picture}}
\vskip -130pt
\caption{
\label{Wilhelm}}
\end{wrapfigure}
\hskip 220pt
\begin{parbox}{500pt}
{\small \baselineskip 11pt 
\vskip -12pt
--- One man asks horrible questions!  \\--- Who is that man? \\--- 
He is called Asker-man!}
\vskip 8mm
\end{parbox}

Here I retell some results for %the
holomorphic ackermanns, tetration and pentation \cite{pentation}.
For base $b\!>\!1$, 
tetration $f\!=\!\mathrm{tet}_b$
is solution of equations
\be
f(z\!+\!1)=b^{f(z)}   ~ ~ ~, ~ ~ ~ f(0)=1
\eL{tetb}
holomorphic at least in range
\be
z \in \bC : \Re(z)>-2 
\eL{htetb1}
that is limited in range
\be
z \in \bC : |\Re(z)| \le 1 
\eL{htetb12}

Equation \rf{tetb} appears as special case of the 
Ackermann equations
\be
\!A_1(z)&\!=\!&b+z ~,~z \in \bC ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~ ~~~~ \iL{Acker1}\\
\!A_n(1)&\!=\!&b ~,~ n\in \bN, n\!\ge\!1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~ ~~~~\iL{Acker2}\\
\!\!\!\!A_n(z\!+\!1)\!\!&\!=\!&\!A_{n-1}\big(A_n(z) \big) ~,~ n\in \mathbb{N}, n > 1 ~ ~ \iL{Acker3}~ ~ ~ ~ ~
\ee
I call functions $A$ ``ackermanns'' in order to avoid confusion with 
mathematician Wilhelm Ackermann, shown in figure \ref{Wilhelm}. His last name, to make difference from the name of the function, is written with 
Capital letter  \cite{acker}. Ackermanns $A$ are subject of this chapter. However, for the highest ackernanns, the range of holomorphism 
has
no need to be the same as condition \rf{htetb1} %,\rf{htetb12} 
for tetration. This range, as well as the asymptotic behaviour of highest ackermanns should be specified. I cannot yet provide the general specification; below, I suggest only the first approach to the problem.
\vskip 5pt

For base $b\!=\!2$, the explicit plots of the first four ackermanna are shown in figure \ref{acker2t} with thick lines,
solid, solid, dotted and dashed. 
The thin lines refers to another (and more conventional) system of notations, described in the next section.
\vskip 2mm


\rule{0pt}{494pt}

\begin{figure}[h]
\vskip -500pt
\sx{.69}{\begin{picture}(308,1006) %\put(12,0){\ing{penma}}
\put(0,0){\ing{acker2}}
\put(281,988){\sx{3.001}{$y$}}
\put(279,895){\sx{3.001}{$6$}}
\put(279,795){\sx{3.001}{$5$}}
\put(279,694){\sx{3.001}{$4$}}
\put(279,594){\sx{3.001}{$3$}}
\put(279,468){\sx{3.001}{$\mathrm e$}}
\put(279,494){\sx{3.001}{$2$}}
\put(279,394){\sx{3.001}{$1$}}
\put(279,294){\sx{3.001}{$0$}}
\put(258,193){\sx{3.001}{$-1$}}
\put(258, 93){\sx{3.001}{$-2$}}
\put( 80,274){\sx{3.001}{$-2$}}
\put(180,274){\sx{3.001}{$-1$}}
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\put(586,274){\sx{3.001}{$x$}}
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\put(348,712){\sx{1.8}{\rot{63}$y\!=\!\mathcal{A}(2,x)\!=\! A_{2,2}(x\!+\!3)\!-\!3$\ero}}
\put(314,526){\sx{1.8}{\rot{45}$y\!=\!\mathcal{A}(1,x)\!=\! A_{2,1}(x\!+\!3)\!-\!3$\ero}}

\put(526,822){\sx{1.8}{\rot{82}$y\!=\!A_{2,4}(x)\!=\!\mathrm{tet}_2(x)$\ero}}
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\put(32,326){\sx{1.9}{\rot{3}$y\!=\!2^x$\ero}}
\put(132,4){\sx{1.9}{\rot{81}$y\!=\!\mathrm{tet}_2(x)$\ero}}
\put(178,8){\sx{1.9}{\rot{66}$y\!=\!\mathrm{2} x$\ero}}
\end{picture}}
\vskip -4pt \hskip 1pt \url{https://mizugadro.mydns.jp/t/index.php/File:Acker2t400.jpg}
\vskip -10pt
\caption{
Comparison of definitions for binary ackermanns
\iL{acker2t}}
\vskip -12pt
\end{figure}

\section{Binary ackermanns}
\baselineskip 18pt

In century 20, the ackermanns are considered mainly  (or even exclusively)  for the base $b\!=\!2$, and mainly (or even exclusively) for integer values of the argument. I call ackermanns to base $b\!=\!2$ as ``binary'' ackermanns.
\vskip 2mm

For the binary ackermanns, the special notations are used. The number of ackermann is written as additional, first argument, and the base $b$ is not indicated at all; there is no need to indicate it, as it remains to be two.
Relation of the classical (and usual for today) notations for the ackermanns by
\rf{Acker1}
\rf{Acker2}
\rf{Acker3} 
can be expressed with the simple formula
\be
\mathcal{A}(m,z)=A_{2,m}(z+3)-3
\eL{asa3}
In particular,
\be
A_{2,1}(x)&=\!&2+x=\mathcal A(1,x\!+\!3)-3=\mathcal A(1,x) \\
A_{2,2}(x)&=\!& ~2\, x ~=\mathcal A(2,x\!+\!3)-3 \\
A_{2,3}(x)&=\!&\exp_2(x)\!=\mathcal A(3,x\!+\!3)-3=2^x \\
A_{2,4}(x)&=\!&\mathrm{tet}_2(x)=\mathcal A(3,x\!+\!3)-3
\ee
\vskip 2mm

Figure  \ref{acker2t} shows relation between the single-argument function $A$ 
with subscripts %and single argument 
and conventional
$\mathcal A$ 
without subscript, but with two arguments.
Four ackermanns are plotted as functions of real argument;
$y\!=\!A_{2,m}(x)~$ for $~m=1,2,3,4$ 
are shown with thick lines (solid, solid, dotted and dashed);
dependences
$y\!=\!\mathcal{A}(m,x)~$ for $~m=1,2,3,4$ 
are shown with thin lines (solid, solid, dotted and dashed);
\vskip 2mm

Due to relation \rf{asa3}, the thick lines in Fig.\ref{acker2t} can be 
obtained
from thick lines by translation for 3 units along $x$ axis and for the same along the ordinate axis; the only straight line for the First ackermann stays at its place at such a transform.
\vskip 2mm

Especially for the binary ackermanns, the system of equations
\rf{Acker1},
\rf{Acker2},
\rf{Acker3}
can be a little but extended, in order to include the ``zeroth" ackermann. Equations for  $\mathcal A$ can be written as follows:
\be
\!\!\!\!\!\!\mathcal A(0,z)&=&z+1 \\
\!\!\!\!\!\!\mathcal A(m\!+\!1,0)&=&\mathcal A(m,1)\\
\!\!\!\!\!\!\mathcal A(m\!+\!1,z\!+\!1)&=&\!\mathcal A\Big(m,\mathcal A(m\!+\!1,z)\Big) ~
\eL{ackerbi}
Displacemenr of both, argument and the function, by formula \rf{asa3} can be qualified as conjugation. The transfer equation \rf{ackerbi} for the binary ackermann in the ``classical'' notations has the same form, as the transfer equation \rf{Acker3}.

\vskip 2mm
One of applications of the ackermanns is to denote the huge numbers (for real moderate values of the argument).
Due to the displacement of the argument, the canonical ackermanns can make illusion of a little bit more fast growth, than the ackermanns by 
\rf{Acker1},
\rf{Acker2},
\rf{Acker3}. 
However, for the applications, this ``acceleration'' is not important. I think, the notations in formulas 
\rf{Acker1},\rf{Acker2},\rf{Acker3} 
are better than the canonical notations.
In notations
\rf{Acker1},
\rf{Acker2},\rf{Acker3},
the third ackermann happens to be just exponent, the fourth - just tetration, and so on.
In addition, I consider ackrmanns as superfunctions, holomorphic with respect to the last argument; so, it is good, to keep this argument single. The base $b$
and the number of the ackermann appear as parameters; they can be specified in the subscript.
I hope, the Reader will meet no problems using relation  \rf{asa3} for the conversion from one system of notations to another one.

\section{Names and notations}

As I had mentioned above, in century 20, the functions of Ackermanns are considered for base  $b\!=\!2$ and only for integer values of the argument $z$.
While, I see no fundamental limits, that would prohibit existence and evaluation of these functions for various, including complex, 
values of $b$ and $z$; however, the appropriate requirement on the range of holomorphism should be formulated.

I hope, for all ackermanns, we may require the real holomorphism at least in some vicinity of at least some part to the real axis.
In general, the solution of the transfer equation is not unique; so, we should indicate also the way of construction of each ackermann, or to guess
(invent, postulate) its behaviour in the complex plane, following the idea, used to construct natural tetration described in chapter 14.
\vskip 6pt

Several ackermanns already have special names:

$A_{b,1}~=~ z\mapsto b\!+\!z$, 
addition of constant
$b$,

$A_{b,2} ~=~ z\mapsto b z$, 
multiplication by constant
$b$,

$A_{b,3}~=~ \exp_b ~=~ z\!\mapsto\! \mathrm{pow}(b,z)~=~z\!\mapsto\! b^z$, 
exponent to base 
$b$,

$A_{b,4}~= ~\mathrm{tet}_b~=~ z \mapsto \mathrm{tet}_b(z)$, 
tetration to base
$b$,

$A_{b,5}~=~ \mathrm{pen}_b~=~ z \mapsto \mathrm{pen}_b(z)$, 
pentation to base
$b$.
\vskip 2mm

The following functions can be denoted with symbols
$\mathrm{hex}_b$,  $\mathrm{hep}_b$, $\mathrm{oct}_b$,..
and be called, accordingly, with terms
``hexation'', ``heptation'', ``octation''
and so on.
These notations appear at the truncation of Latin names of numbers to three characters.
\vskip 6pt

Initially, the formalism of sulerfunctions had been developed for tetration, id est, for superfunction of the exponent, for the Fourth ackermann.
However, this formalism can be applied also for other functions, and, in particular, for various ackermanns.
As an example, in the next section I consider the 5th ackermann, the pentation.

\section{Pentation}

If you give to some true Mathematician the teapot, the gas stove, the matches and water tap, and ask 
to prepare tea, the Mathematician puts water into the teapot, ignites the gas and put the teapot on the fire.
If the Mathematician finds the tea powder, then, perhaps, he or she even drop it into the hot water after to see the boiling and switch off the gas.
But if, after that, the Mathematician again is asked to prepare the tea, while the water is already in the tea pot, and the gas fire at the stove is ready..
Ooh.. You may guess, that the true Mathematician shuts down the gas, drops the water from the teapot and spells the magic sentence:
{\bf ``The problem is reduced to the previous one!''}.
\vskip 6pt

I hope, with methods, described above, and keeping in mind the previous paragraph, 
the Reader already can calculate the superfunctions, just following the general algorithms from this Book.
On the other hand, I am more physicists, than mathematician. 
(The mathematician colleagues have no doubt in this, although some physicists express the opposite opinion.)
Therefore, instead of to spell the magic sentence {\bf ``The problem is reduced to the previous one!''},
in this chapter, I suggest one more example of evaluation of ackermann.
This example refers to the natural pentation, id est,
$\mathrm{pen}=\mathrm{pen}_\mathrm{e}=A_{\mathrm e,5}$.
\vskip 6pt

\begin{figure}[t]
\normalsize
\vskip -2pt
\sx{.6}{\begin{picture}(702,702)\put(0,0){\ing{tet5loplo}}
\put( 172,680){\sx{4}{$y$}}
\put( 172,590){\sx{4}{$4$}}
\put( 182,467){\sx{4}{e}}
\put( 172,390){\sx{4}{$2$}}
\put( 172,190){\sx{4}{$0$}}
\put( 194,164){\sx{4}{$0$}}
\put( 394,164){\sx{4}{$2$}}
\put( 594,164){\sx{4}{$4$}}
\put( 686,166){\sx{4}{$x$}}
\put(344,570){\sx{4}{\rot{80} $b\!=\! \rme$ \ero } }
\put(522,574){\sx{4}{\rot{66} $b\!=\!1.7$ \ero } }
\put(600,603){\sx{4}{\rot{64} $b\!=\!\tau$ \ero } }
\put(550,492){\sx{4}{\rot{42} $b\!=\!1.6$ \ero } }
\put(580,438){\sx{4}{\rot{14} $b\!=\!1.5$ \ero } }
\put(592,400){\sx{4}{\rot{7} $b\!=\! \eta$ \ero}}
\put(574,345){\sx{4}{\rot{2}$b\!=\!\sqrt{2}$\ero}}
\put(130,504){\sx{4}{$L_{\tau,1}$}}
\put(500,164){\sx{4}{$L_{\tau,1}$}}
\put(-2,219){\sx{4}{$L_{\mathrm e,0}$}}
\put(202,6){\sx{4}{$L_{\mathrm e,0}$}}
\end{picture}}
\vskip 1pt
{\url{https://mizugadro.mydns.jp/t/index.php/File:Tet5loplot.jpg}}
\vskip -4pt
\caption{$y\!=\!\mathrm{tet}_b(x)$, 
fragment of figure
\ref{tet10bx}; 
line
$y\!=\!x$ 
and 
$y\!=\!\mathrm{tet}_\tau(x)$
are added
 \label{tet5loplot}}
\vskip -8pt
\end{figure}

At the construction of a superfunction, the key question is about the fixed points of the transfer function.
For pentation, the transfer function is tetration, considered in chapter 14.
The real fixed points of tetration are shown in figure \ref{tet5loplot}. This is modification of figure \ref{tet10bx}:
some lines are removed, and the new curve for tetration to base
$b\!=\!\tau\!\approx\!1.63532$ is added. Here, $\tau$ is specific values of base, at which the curve $y\!=\!\mathrm{tet}_\tau(x)$
touches the line $y\!=\!x$.
The point of touching has coordinates
$(L_{\tau,1},L_{\tau,1})$, 
and
$L_{\tau,1}\approx 3.087$.
For this value, the additional grid lines are added in figure 
\ref{tet5loplot}.
\vskip 6pt

At base $b\!>\!\tau$, tetration  $\mathrm{tet}_b$ has the only one real fixed point. In particular, this refers to the case of natural tetration
$\mathrm{tet}=\mathrm{tet}_\mathrm e$. For natural tetration, this point is 
$L\!=\!L_{\mathrm e,0}\!\approx\! -1.85035452902718$, 
and for this value, also the additional grid lines are shown in figure
\ref{tet5loplot}.
Namely this point is chosen to built-up the natural pentation pen, id est, for the fifth ackermann to base
$\mathrm e\!=\!\exp(1)\!\approx\! 2.71~$.
As the Reader can see, not so many options we have in this case.
\vskip 6pt

The growing real-holomorphic superfunction of natural tetration, that is constructed with regular iteration at the fixed point
$L_{\mathrm e,0}$ %\!\approx\! -1.85035452902718$ 
and approaches this fixed point at $-\infty$, I call the fifth ackermann, or pentation.
Graphic of pentation is shown in figure \ref{penplo}. I describe the construction below.
\newpage
\begin{figure}[t]
\normalsize
\sx{.7}{\begin{picture}(608,908) \put(0,0){\ing{penplo7}}
\put(377,894){\sx{3.2}{$y$}}
\put(377,795){\sx{3.2}{$6$}}
\put(377,695){\sx{3.2}{$5$}}
\put(377,594){\sx{3.2}{$4$}}
\put(377,494){\sx{3.2}{$3$}}
\put(377,466){\sx{3.2}{$\mathrm e$}}
\put(377,394){\sx{3.2}{$2$}}
\put(377,294){\sx{3.2}{$1$}}
\put(377,194){\sx{3.2}{$0$}}
\put(358, 93){\sx{3.2}{$-1$}}
\put(80,174){\sx{3.2}{$-3$}}
\put(180,174){\sx{3.2}{$-2$}}
\put(280,174){\sx{3.2}{$-1$}}
\put(396,174){\sx{3.2}{$0$}}
\put(496,174){\sx{3.2}{$1$}}
\put(590,174){\sx{3.2}{$x$}}
\put(242,406){\sx{3.6}{\rot{85}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}}
\put(534,472){\sx{4}{\rot{75}$y\!=\!\mathrm{pen}(x)$\ero}}
\put(8,236){\sx{3.3}{$y=10\,\delta(x)$}}
\put(252, 10){\sx{3.2}{$y\!=\!L_{\mathrm e,4,0}$}}
\end{picture}}
\vskip 2pt {\url{https://mizugadro.mydns.jp/t/index.php/File:Penplot.jpg}}
\vskip 2pt
\caption{
$y\!=\!\mathrm{pen}(x)$ 
by
\rf{pendef}, 
its asymptotic
\rf{pen01e},
and deviation of the linear approximation by
\rf{pendelta}
\label{penplo}}
\vskip -400pt
\end{figure}
~
\newpage
\baselineskip 17.6pt


For superfunction $F$ of transfer function tet, the transfer equation is
\be\!\!\!\!
F(z\!+\!1)=\mathrm{tet}\Big( F(z)\Big)
\label{TranPen}
\ee
I construct the growing along the real axis solution $F$ by the regular iteration at the
fixed point of tetration
$L\!=\!L_{\mathrm e,5,0}\!\approx\! -1.85035452902718$; 
I mention the key point of the construction below.
\vskip 6pt

For some natural number $M\!>\!1$, I search the solution  $F$ of equation \rf{TranPen} in the following form:
\be
F(z)=f(z) + O(\varepsilon^M)
\eL{penF}
where
\be
f(z)&=&L_{\mathrm e,4,0}+\sum_{m=1}^{M-1} a_m \varepsilon^m   ~ ~ ~ \iL{penf}
\\
\varepsilon&=&\exp(k z) ~ ~ ~ ~ ~ ~
\eL{penepsilon}
Here, the positive constant 
$k$ has sense of the increment of the growth of superfunction at large negative values of the argument,
and $a$ are real coefficients.
For simplicity, I set
$a_1\!=\!1$. 
Substitution of representations
\rf{penF}, \rf{penf}
to the transfer equation
\rf{TranPen} 
and the asymptotic analysis with small parameter
$\varepsilon$ 
give
\be
k=\ln\big(\mathrm{tet}'(L)\big) \approx 1.865733
\label{kpen}
\ee
and the coefficients $a$; in particular,
\be
a_2 =\!&\!\displaystyle \frac{\mathrm{tet}''(L)/2}{\Big(\mathrm{tet}'(L)\!-\!1\Big)\mathrm{tet}'(L)}     \!&\!\approx -0.6263241 \\
a_3 =\!&\!  \displaystyle \frac{\mathrm{tet}''(L)a_2+\mathrm{tet}'''(L)}{\Big(\mathrm{tet}'(L)^2\!-\!1\Big)\mathrm{tet}'(L)}   \!&\!\approx ~ ~  0.4827
\ee
\baselineskip 20pt
For the numerical implementation, in \rf{penf}, I choose $M\!=\!4$; this is sufficient to evaluate pentation with 14 significant figures and to plot all the figures of this article in real time.
This approximation is good for large negative values of the real part of argument of supertetration.
Then, for integer $n$, I define
\be
F_n(z)=\mathrm{tet}^n(f(z-n))
\eL{pentalim}
The exact superfunction $F$ appears as limit
\be \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!
F(z)=\lim_{n\rightarrow \infty} F_n(z)
\eL{flim}
The limit does not depend on the chosen number $M$ of terms in the asymptotic expansion.
However, the larger is $M$, the faster the limit in \rf{flim} does converge.
\vskip 2mm

The pentation appears as superfunction $F$ with displaced argument:
\be
\mathrm{pen}(z)=A_{\mathrm e,4}(z)=F(x_5\!+\!z)
\eL{pendef}
where $x_5\!\approx\!2.24817%45%1898
$ is solution of equation $F(x_5)\!=\!1$.
Complex map of this pentation is shown in figure \ref{penmap}.%, announced above.


\begin{figure}[t]
\normalsize
\begin{center}
\sx{.5}{\begin{picture}(824,820)
\put(12,0){\ing{penma}}
\put(8,808){\sx{3}{$y$}}
\put(8,709){\sx{3}{$3$}}
\put(8,609){\sx{3}{$2$}}
\put(8,509){\sx{3}{$1$}}
\put(8,409){\sx{3}{$0$}}
\put(-12,309){\sx{3}{$-1$}}
\put(-12,209){\sx{3}{$-2$}}
\put(-12,109){\sx{3}{$-3$}}
\put(-12,9){\sx{3}{$-4$}}
\put(4,-8){\sx{3}{$-4$}}
\put(104,-8){\sx{3}{$-3$}}
\put(204,-8){\sx{3}{$-2$}}
\put(304,-8){\sx{3}{$-1$}}
\put(427,-8){\sx{3}{$0$}}
\put(527,-8){\sx{3}{$1$}}
\put(627,-8){\sx{3}{$2$}}
\put(727,-8){\sx{3}{$3$}}
\put(821,-8){\sx{3}{$x$}}
\put(36, 747){\sx{4}{$v\!=\!0$}}
\put(30, 664){\sx{3.3}{\rot{-4}$u\!=\!L$\ero}}
\put(36, 578){\sx{4}{$v\!=\!0$}} \put(760, 580){\sx{4}{\bf cut}}
\put(30, 498){\sx{3.3}{\rot{2}$u\!=\!L$\ero}}
\put(36, 409){\sx{4}{$v\!=\!0$}}%
\put(30, 324){\sx{3.3}{\rot{-2}$u\!=\!L$\ero}}
\put(36, 240){\sx{4}{$v\!=\!0$}} \put(760, 241){\sx{4}{\bf cut}}
\put(30, 158){\sx{3.3}{\rot{2}$u\!=\!L$\ero}}
\put(36, 71){\sx{4}{$v\!=\!0$}}
\put(326, 638){\sx{4}{$v\!=\!-1$}}
\put(340, 520){\sx{4}{$v\!=\!1$}}
\put(326, 298){\sx{4}{$v\!=\!-1$}}
\put(336, 182){\sx{4}{$v\!=\!1$}}
\put(250, 352){\sx{4}{\rot{90}$u\!=\!-1$\ero}}
\put(348, 362){\sx{4}{\rot{90}$u\!=\!0$\ero}}
\put(448, 372){\sx{4}{\rot{90}$u\!=\!1$\ero}}
\put(522, 372){\sx{4}{\rot{90}$u\!=\!2$\ero}}
\end{picture}}
\end{center}
\vskip -9pt
\hskip 36pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Penmap.jpg}
\vskip -2pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{pen}(x\!+\!\mathrm i y)$ по формуле \rf{pendef} \iL{penmap}}
\vskip 18pt
\end{figure}
\vskip 6pt

The real-real plot of pentation by \rf{pendef} is shown in figure  \ref{penplo} with thick curve. The additional horizontal grid line
$y\!=\!L_{\mathrm e,4,0}$ shows the asymptotic at large negative values of the argument. The thin curve shows the 
more advanced
asymptotic
\be
y\!=\!L_{\mathrm e, 4,0}+\exp(k(x\!+\!x_5))
\eL{pen01e}

Pentation is holomorphic at least for $|\Re(z)|\!<\!|P|/2\approx 1.6838%38
$,
where $P\!=\!2\pi\mathrm i /k $ is period; pentation, as exponential, is periodic function.
A little bit more than two periods are covered by the range of the map at figure 
\ref{penmap}.
Pentation has the countable set of cut lines, parallel to the real axis. In figure \ref{penmap},
these cuts are marked with dashed lines.

\vskip 6pt

\JP{ DUPLICATE!!!
In figure \ref{penplo}, the additional horizontal line line indicates ordinate $L\!=\!L_{\mathrm e,4,0}$, 
this line is borrowed from figure
\ref{tet5loplot}. 
It indicates the limiting value of pentation at minus infinity; this is horizontal asymptote of the graphic of pentation.
In addition, the thin line shows the exponential (id est, a little bit more advanced) asymptotic
\be
y=L_{\mathrm e,4,0}+\exp(k (x\!+\!x_1))
\label{pen01e}
\ee
This asymptotic corresponds to value $M\!=\!2$ in the primary approximation of dentition by formula \rf{penf}.
\vskip 2mm}

In vicinity of the segment of length 2 at the negative part of the real axis,
pentation can be approximated with the linear function,
\be
\mathrm{pen}(x)\approx 1+x
\eL{penlin}
At $-2.1\!<\! x \! <\! 0.1$, approximation \rf{penlin} provides two significant figures. 
Deviation
of this approximation from pentation pen can be expressed with function
\be \!\!\!\!\!\!\!\!
\delta(x)=\mathrm{pen}(x) -(1\!+\!x) ~
\label{pendelta}
\ee
This 
deviation
is shown in figure \ref{penplo} with thin line;
it is scaled with factor 10;
curve 
$y=10\, \delta(x)$
is drawn.
\vskip 2mm

The linear function in the right hand side of equation \rf{penlin} 
approximates also the previous ackermann, id est, tetration; its graphic is show in figure
\ref{TetPlot}.
For tetration, the function in the right hand side of formula
\rf{penlin} also gives of order of two significant figures;
however, the range of this approximation for pentation is twice wider, than for tetration.
\vskip 2mm

Complex map of pentation by
\rf{pendef} 
in figure \ref{penmap} demonstrates, that
$\mathrm{pen}(z)$ is holomorphic at least for $\Re(z)<-2.5\,$. As the real part of the argument approaches minus infinity,
pentation exponentially approaches the limiting case
$L\!=\!L_{\mathrm e,4,0}\!\approx\! -1.850354529$,
shown in figure 
\ref{tet5loplot}.
In order to show this explicitly, the light strip in figure 
\ref{penmap} 
indicates the additional level
$u\!=\!L_{\mathrm e,4,0}$.
\vskip 2mm

Pentation is periodic; its period $P$ is determined by the increment $k$, 
id est, by the derivative of tetration at its fixed point
$L_{\mathrm e,4,0}$:
\be
P=\frac{2 \, \pi \, \mathrm i}{k} \approx 3.36767615657879 \, \rm i 
\eL{penP}
The period is pure imaginary;
the complex map of pentation reproduces itself at translations for integer factor of the period.
The cuts of the range of holomorphism are also reproduced.
\vskip 2mm
Along the real axis, pentation shows very fast growth, faster, than that of natural tetration.
As maps of many other fast-growing functions, the map of pentation has complicated structure in vicinity of the real axis.
Pentation varies with huge derivatives, that correspond to enormous density of the levels at the complex map. The plotter could not draw the huge amount of lines, and so, the narrow region in vicinity of the real axis in figure
\ref{penmap} is left empty. The same applies to the translations for integer factor of period $P$.
In order to shown behaviour of pentation in vicinity of the real axis and in vicinity of the cut line, figure \ref{penzoo} shows the zoom-in from the figure \ref{penmap}.
\vskip 2mm


General methods of construction of superfunctions can be used to build-up the ackermanns.
If the growth of tetration happens to be not sufficiently fast, the pentation described above, 
can be used. The highest ackermanns can be constructed in the similar way.
In particular, the fixed points of pentation indicate the way to build-up its superfunction, id est, the next ackermann.
Fixed points of tetration are considered in the next section.

\begin{figure}
\sx{.82}{\begin{picture}(512,208)
\put(8,9){\ing{penzoo}}
\put(1,205){\sx{1.2}{$y$}}
\put(-.4,176.8){\sx{1.1}{$\frac{P}{2 \rm i}$}}
\put(130,177){\sx{1.2}{\bf cut}}
\put(1,158){\sx{1.1}{$\frac{3}{2}$}}
\put(1,107){\sx{1.2}{$1$}}
\put(1,057){\sx{1.17}{$\frac{1}{2}$}}
\put(1,7){\sx{1.2}{$0$}}
\put(50,0.4){\sx{1.12}{$-2$}}
\put(150,0.4){\sx{1.12}{$-1$}}
\put(258,0.5){\sx{1.12}{$0$}}
\put(358.4,0.5){\sx{1.12}{$1$}}
\put(459,0.5){\sx{1.12}{$2$}}
\put(505,0.5){\sx{1.12}{$x$}}
\put(392,73){\sx{1.7}{\rot{61}$v\!=\!2$\ero}}
\put(212,120){\sx{1.7}{\rot{0}$v\!=\!1$\ero}}
\put(195.6,5.8){\sx{1.7}{\rot{0}$v\!=\!0$\ero}}
\put(063,17){\sx{1.7}{\rot{78}$u\!=\!-1$\ero}}
\put(166,18){\sx{1.7}{\rot{90}$u\!=\!0$\ero}}
\put(265,18){\sx{1.7}{\rot{81}$u\!=\!1$\ero}} 
\put(334,15){\sx{1.7}{\rot{65}$u\!=\!2$\ero}}
\put(29.5,146){\sx{1.2}{$L$}}
\put(361.4,162.3){\sx{1.2}{$L$}}
\end{picture}}
\vskip -2pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Penzoo25t400.jpg}
\vskip -6pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{pen}(x\!+\!\mathrm i y)$ 
by \rf{pendef}, 
zoom-in from figure
\ref{penmap} \iL{penzoo}}
\vskip 1pt
\end{figure}
\section{Fixed poins of pentation and future work}

In this section I suggest some hint, how can one built-up the next ackermann, hexation, using the fixed points of pentation.
Actually, here I do not construct hexation (because, anyway, I have to stop somewhere); I only mention the way to do it.
\vskip 2mm

As usually, one should begin with the fixed points, id est, solutions $L$ of  
\be
\mathrm {pen}(L)=L
\eL{penLeq}
Some of the solutions are shown in figure \ref{penzoo}, these solutions are marked with character $L$. They correspond to
\be
L&=&L_{\mathrm e, 5,0}\approx -2.260+1.384 \, \mathrm i \\
L&=&L_{\mathrm e, 5,1}\approx ~ 1.057+1.546 \, \mathrm i
\eL{penL1}
There are also solutions in vicinity of the real axis
\be
L&=&L_{\mathrm e, 5,2}\approx 3.43+0.07 \, \mathrm i \\
L&=&L_{\mathrm e, 5,3}\approx 4.39+0.11 \, \mathrm i
\eL{penL3}
but they do not fit the frame of the map on figure \ref{penzoo}.
The readers are invited to solve numerically equation \rf{penLeq}
and to adjust values of
$L_{\mathrm e, 5,0}$ 
$L_{\mathrm e, 5,1}$.
I expect, one can find the real-holomorphic solution $F$ to equation
\be
F(z\!+\!1)=\mathrm{pen}\Big( F(z) \Big)
\ee
with additional conditions
\be
F(0)\!=\!1 ~,~~
F(\mathrm i \infty) \!=\! L_{\mathrm e, 5,0} ~,~~
\ee
The readers are invited to find this solution and interpret it as hexation, id est, the 6th ackermann.
\vskip 8pt

At this point, I stop constructing new ackermanns. I want to compare the first 5 ackermanns. It is mater of the next section.


\newpage

\section{Comparison of natural ackermanns}
\baselineskip 19pt
In this section I compare the first 5 ackermanns for the natural base $b\!=\!\mathrm e$. In such a way, I overview results for ackermanns.
For real values of argument, these functions are plotted in figure 
\ref{ackerplo}. 
\vskip 3mm

\begin{figure}[h]
\normalsize
\vskip  4pt
\sx{.7}{\begin{picture}(608,808) %\put(12,0){\ing{penma}}
\put(0,0){\ing{ackerplo}}
\put(277,788){\sx{3.001}{$y$}}
\put(277,695){\sx{3.001}{$5$}}
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\put(308, 13){\sx{2.2}{$y\!=\!L_{\mathrm e,4,0}$}}
\end{picture}}
\vskip -4pt
\hskip 4pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Ackerplot400.jpg}
\vskip -4pt
\caption{\normalsize $y\!=\!A_{\mathrm e,n}(x)$ for $n\!=\!1,2,3,4,5$ \iL{ackerplo}}
\vskip -400pt
\end{figure}
\newpage
Functions plotted in figure \ref{ackerplo} are:\\
$A_{\mathrm e,1}(z) = \mathrm e\!+\!z$, 
addition of constant
$\mathrm e$,\\
$A_{\mathrm e,2}(z) = \mathrm e\, z$, 
multiplication to constant
$\mathrm e$,\\
$A_{\mathrm e,3}(z)= \exp(z)= \mathrm e^z$, 
natural exponent
,\\
$A_{\mathrm e,4}(z)=\mathrm{tet}(z)$, 
tetration to base 
$\mathrm e$,\\
$A_{\mathrm e,5}(z)=\mathrm{pen}(z)$, 
pentation to base
$\mathrm e$.
\vskip 1mm

These notations are used to mark curves
$y\!=\!A_{\mathrm e,n}(x)$ 
at figure
\ref{ackerplo}. 
As usually, if the base is not specified at the subscript, it is assumed to be $\mathrm e$, base of natural exponent and that of natural logarithm.

\vskip 2mm
Tetration and pentation at the segment $[-1,1]$ look similar. However, at the printing with good resolution,
the deviation is seen, it exceeds the width of the lines in figure 
\ref{ackerplo}.
\vskip 2mm

I expect, the tetration already has the growth fast enough to describe the fastest function that may 
appear in the applications. However, the main property of the scientific revolution is that they are unexpected.
If for some case, the growth of tetration is not fast enough, the pentation or even higher sckermanns can be implemented, using the tools from this Book.
\vskip 4mm


When I plotted the pentation, I ask myself: "Why not to do in the similar way the next ackermann?"
The answer is simple: I already know, how to do it. And the Reader, if reached here, also knows.
Anyway, the Book should be finished at some moment, see figure \ref{veslo}.
\vskip 2mm

Instead of to add more examples for the tools described above, I think, it is more important to consider at least one example, when the tools above do not work. This example correspond to the transfer function, that has no fixed points at all - neither real, nor even complex.
Such an example is considered in the next chapter, and, while typing this, I believe, that will be the last example in this Book.

\JP{
\section{Last coefficient sindrom}
Я спрашиваю себя: ``{\em почему бы не сделать гексацию так же, как уже сделана тетрация и пентация?''}. Для того, чтобы объяснить это, мне прийдется копипастнуть ещё одно лирическое отступление. 
\vskip 3mm

Допустим, исследователь придумал совершенно новую и безумно красивую функцию. Он строит график, ну, скажем, от $-4$ до $4$. Не все пышные округлости функции умещаются на графике, и исследователь строит её же от $-10$ до $10$. Но последняя осцилляция не видна, и он расширяет область значений до интервала $[-11,11]$. И так далее. Наконец, когда график расширился настолько, что центральную часть уже видно только в лупу, исследователь начинает вычислять, ну, скажем, её разложение в ряд Тэйлора. Он считает первые 5 коэффициентов и вклеивает их в свою фундаментальную статью. Он смотрит на них, и ему кажется, что следующий коэффициент мог бы помочь пониманию. Он считает шестой коэффициент и расширяет таблицу. Но ему тут же кажется, что еще один (только один!) коэффициент сильно улучшит статью...
\vskip 1mm

- Стоп, - оборвет меня Читатель, - Что за чушь? Он что, больной?.. 
\vskip 1mm

Да, дорогой Читатель, да; именно так. Эта болезнь называется Синдром Последнего Коэффициента. Примерно так же в повести о Хадже Насреддине ростовщик Джафар каждый раз мечтает наполнить золотом еще один (только один!) горшок 
\footnote{ \url{https://coollib.net/b/43714/read} Леонид Соловьев. Возмутитель спокойствия. (Повесть о Хадже Насреддине) ..во дворе собрались многочисленные родственники Джафара. Многие из них были должны ему и надеялись, что сегодня на радостях он простит долги. Напрасно они надеялись: ростовщик слышал через закрытую дверь голоса своих должников и злобно усмехался в душе. «Я скажу им сегодня, что прощаю долги, – думал он, – но расписок я им не верну, расписки останутся у меня. И они, успокоившись, начнут беспечную жизнь, а я ничего не скажу, я буду молчать, но втайне все время подсчитывать. И когда на каждую таньга долга нарастет еще десять таньга и сумма долга превысит стоимость домов, садов и виноградников, принадлежащих ныне моим должникам, я позову судью, откажусь от своего обещания, предъявлю расписки, продам все их имущество, оставлю их нищими и наполню золотом еще один горшок!»}. Примерно так же маньяк-убийца, получив высшую власть, каждый раз думает, что теперь достаточно ``убрать'' (так на чекистском жаргоне называется политическое убийство) пару кляузников (всего двух!), и стране настанет полный порядок. Или как алкоголик каждый раз хочет только одну (единственную!) рюмку водки. 
\vskip 1mm

Все упомянутые лица больны синдромом последнего коэффициента. Я тоже страдаю от этого недуга. В порядке лечения, я не включаю сюда гексацию, то есть шестого аскерманна, ни даже аркпентацию, которая нужна для построения шестого аскерманна методом Коши - тем же методом, которым строится натуральная (и не только) тетрация. 
\vskip 2mm

Сколько бы аскерманнов я не загрузил, всегда будет такой, для которого  в Книге ещё нет ни одного рисунка. Есть и иная мотивация к тому, чтобы в какой-то момент остановиться с этой Книгой - чтобы все-таки закончить Книгу в этом веке. Эта мотивация показана на рисунке \ref{veslo}.
\vskip 4mm


На этом я заканчиваю рассмотрение регулярных и не очень итраций, а также прочих интегралов Коши. Надеюсь, что читатель уже понял, чем Фудоки отличаются от Нибелунгов,  
и может самостоятельно построить всех аскерманнов, которые ему понадобятся.
\vskip 3mm

Теперь гораздо важнее рассмотреть хотя бы один случай, когда методы, описанные выше, не работают. Такое очень даже может приключиться, если у трансцендентной передаточной функции вообще нет стационарных точек. Пример такой функции рассмотрен в следующей главе.
}
 %\JP{

\begin{figure}[b]
\sx{.35}{\begin{picture}(500,330)
\put(140,-10){\ing{veslo}}
\put(377,299){\color{white}\rule{400pt}{160pt}}
\put(390,350){\sx{3}{No rafting! Sit and finish your divine}}
\put(390,310){\sx{3}{Book ``Superfunctions''}}
\put(100,420){\sx{2.8}{\url{https://mizugadro.mydns.jp/t/index.php/File:Veslo.png}}}
\end{picture}}
\vskip -18pt
\caption{%\large  %Девушка с веслом 
Wash yourself and finish your divine opera {\em Khovanshina}!
\cite{erofeev} 
\label{veslo} ~ ~ ~ ~}
\vskip -12pt
\end{figure}



\chapter{~ Without fixed points}
\baselineskip 17.6pt

In this chapter I describe the last (for this book) example of the transfer function, I consider function without fixed points. I suggest, that the Reader tries to invent, to guess, to write-out  some entire function without fixed points, before to look at the formula below.
\vskip 2mm

For the transfer function without fixed points, neither method of regular iteration, nor the representation through the Cauchy integral can be applied as is for the primary evaluation of superfunction.
For this reason, this example is interesting and deserves the special chapter.
Here, I retell results published recently \cite{sutra}.

\begin{wrapfigure}{r}{216px}
\vskip -18pt
\normalsize
\sx{.41}{\begin{picture}(604,660)
\put(0,0){\ing{traplot}}
\put(280,784){\sx{3.4}{$y$}}
\put(280,693){\sx{3.4}{$4$}}
\put(224,666){\sx{3.4}{$1\!+\!\rm e$}}
\put(280,593){\sx{3.4}{$3$}}
\put(280,566){\sx{3.4}{$\rm e$}}
\put(280,493){\sx{3.4}{$2$}}
\put(280,393){\sx{3.4}{$1$}}
\put(280,294){\sx{3.4}{$0$}}
\put(252,191){\sx{3.4}{$-1$}}
\put(252,091){\sx{3.4}{$-2$}}
\put(070,272){\sx{3.4}{$-2$}}
\put(170,272){\sx{3.4}{$-1$}}
\put(294,272){\sx{3.4}{$0$}}
\put(394,272){\sx{3.4}{$1$}}
\put(482,274){\sx{3.4}{$x$}}
\put(24,342){\sx{3.4}{$y\!=\!\exp(x)$}}
\put(56,42){\sx{3.4}{$y\!=\!\mathrm{tra}(x)$}}
\put(534,0){\sx{2.33}{\rot{90}\url{https://mizugadro.mydns.jp/t/index.php/File:TraplotT.jpg}\ero}}
\end{picture}}
\vskip -10pt
\caption{\large
$y\!=\!\mathrm{tra}(x)$ 
and
$y\!=\!\exp(x)$
\label{traplot}}
\vskip -24pt
\end{wrapfigure}

\section{Trappmann function}

I give name `Trappmann function'' to the elementary function
\be \!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{tra}(z)=\exp(z)+z
\label{tra}
\ee
Function tra has no fixed point; equation
$\mathrm{tra}(L)\!=\!L$ has no solution.
Function tra is compared to exponent in figure 
\ref{traplot}.
\vskip 6pt

Function tra had been expected to be a trap, trump, to catch me on my 
claim, that I can build-up
a superfunction for any growing real-holomorphic transfer function.
Henryk Trappmann had suggested this function; so, I use first three caracters of his family name to denote it.
\vskip 6pt

Word ``trap'' can be interpreted also as gate, a way to the future successes in solving of various transfer equations and iterates of tricky functions.
We see, function tra
deserves the special name.
\vskip 6pt


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\vskip -2pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:TraMapT.jpg}
\vskip -6pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{tra}(x\!+\!\mathrm i y)$ по формуле \rf{tra} \iL{tramap}}
\vskip 2pt
\end{figure}
Complex map of function tra by \rf{tra} is shown in figure \ref{tramap}. 
In the right hand side of the figure, the map looks similar to that go exponent:
at the background of the exponential growth, the linear addition in the right hand side of equation \rf{tra} does not look impressive.
In the left hand side of the map, contrary, the exponent becomes negligible, and the lines of constant real part and those of constant imaginary part of the function form almost uniform grid of lines, parallel to the coordinate axes.
\vskip 2mm
Similar property, at least in some part of the complex plane, is shown by the inverse function, id est,
$\mathrm{ArcTra}=\mathrm{tra}^{-1}$.
Complex map of ArcTra is shown in figure \ref{arctranmap}.
I consider its evaluation in the next section.


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\vskip -8pt
\hskip 24pt
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\vskip -6pt
\caption{\large $u\!+\!\mathrm i v\!=\!\mathrm{ArcTra}(x\!+\!\mathrm i y)$ ~   \iL{arctranmap}}
\vskip 6pt
\end{figure}


\section{ArcTra, inverse of trappmann}
\baselineskip 19.8pt

In order to iterate a function, I mean non-integer iterates, first, we need to learn to evaluate the integer iterates.
For positive integer number of iterate, we may apply the function to the argument so many times as necessary.
As for the negative iterates, we need the efficient algorithm for the inverse function.
Let this inverse function be called ArcTra, in analogy tithe ArcSin, ArcTan and ArcBessel;
in some wide range to values of the argument (that includes the real axis), the ArcTra should satisfy equation
\be
\mathrm{tra}\Big( \mathrm{ArcTra}(z)\Big)=z
\eL{traArcTra}
Complex map of function ArcTra is shown in figure 
\ref{arctranmap}.
\vskip 2mm
\begin{wrapfigure}{r}{190px}
\vskip -22pt
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\put(0,-12){\ing{optic}}
\put(500,0){\sx{1.8}{\rot{90}\url{File:LinearNonlinearOptic.png} \ero}}
\end{picture}}
\vskip -4pt
\caption{%\large
Linear versus nonlinear
\label{optic}}
\vskip -10pt
\end{wrapfigure}

Perhaps, I should explain, why I describe in this Book so elementary thing, as building-up of inverse function.
In century 20, I used to deal with students interested in nonlinear optics and quantum optics.
Some of them were smart \cite{edua}, but some students had problem even with linear optics.
It costed to me certain efforts, to explain them, that there is a lot of pretty ``linear'' science behind every so-called ``nonlinear'' phenomenon, as it is shown in figure \ref{optic}.
In the similar way, it is vain to discuss non-integer iterate, while even the negative integer iterates are not yet implemented. I describe the implementation of function ArcTra in this section.
\vskip 2mm

During the USSR, there was some science there. The famous institutes of the so-called "Soviet School"
were Fizfak (Физфак, Physics department of the Moscow State University) and 
Fiztech (Физтех, Moscow Phisics-Technical Institute). As one can guess, Fizfak used to deal with fundamental science, and Fiztech did with the applied one. In order to 
compare a
specialist graduated from Fizfak to that graduated from Fiztech, in the USSR, the following example is popular.
One, graduated from Fiztech, can calculate or ensemble everything. But he/she understands close to nothing. As for one graduated from Fizfak - Oooh.. he or she understands everything, although can calculate close to nothing. I remind that story for the analogy with figures of this section. Readers, who are interested in the beautiful pictures, may look at the complex maps presented in this section. Then, these Readers can be qualified with specification ``understand everything''. As those graduated from Fizfak.
\vskip 2mm

\EN{For the implementation of the Abel function for the transfer function tra, the inverse function $\mathrm{ArcTra}=\mathrm{tra}^{-1}$ is used.
Unfortunately, as in the case of ArcSin and ArcFactorial, I could not find any complex double implementation of ArcTra, and I had to make it by myself.
This is general rule: for the efficient implementation of the non-integer iterates of some transfer function 
\footnote{In principle, any holomorphic function can be declared as ``transfer function''. I still specify that tra is ``transfer function'', in order to indicate my intention to iterate it and to distinguish it from its superfunction and its Abel function.} , the inverse function also should be implemented.
In order to show the underwater stones, that may appear at calculation of superfunctions, in this section, the efficient implementation of function ArcTra is described.}
\vskip 2mm

\RU{Как уже отмечено выше, функция tra является элементарной, и не составляет труда сосчитать её производную:}%
It is not difficult to calculate the derivative of function tra:
\be \!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
\mathrm{tra}(z)=z+\exp(z) ~~, ~~~ \mathrm{tra}^{\prime}(z)=1+\exp(z)
~ ~ ~ ~~
\eL{traagain}
\EN{The Newton method 
gives a
good precision evaluating solution $f$ of equation}%
\RU{Для построения функции ArcTra требуется 
решение $f$ уравнения}%
\be
\mathrm{tra}(f)\!=\!z
\eL{traf}
\RU{Последовательность $f_n$ аппроксимаций решения $f$ можно строить методом Ньютона, }
\be
f_{n+1}=f_n + \frac{z-\mathrm{tra}(f_n)}{\mathrm{tra}'(f_n)} ~ ~ ~ ~ ~
\eL{arctranewton}
\RU{при подходящем начальном значении $f_0$. Я считаю, что начальное значение подходит, если в пределе}
\be
f=\lim_{n\rightarrow \infty} f_n
\eL{arctralim}
\RU{решение $f$ оказывается на том листе многолистной функции, который мне нравится.
Как уже отмечено, мне нравится, когда линии разреза идут от точек ветвления прямо в направлении минус бесконечности, параллельно вещественной оси. Именно так идут разрезы на рисунке \ref{arctranmap}. Мелкая сетка, образованная изолиниями в полосе вдоль отрицательной части вещественной оси, то есть оси $x$ на рисунке \ref{arctranmap}, достаточно четко указывает положение разрезов, так что я не стал отмечать их дополнительным пунктиром.
}%
\EN{under a simple condition: the initial approximation $f_0$ indicates the correct branch of the resulting inverse function.
It is general rule, that any non-trivial holomorphic function (and, especially, entire function) has some 
points, where its derivative is zero; and these points provide branches of the inverse function. For this reason, for the robust implementation, the correct choice of the initial approximation is essential: over-vice, the resulting function may return values from different branches in some almost random, almost unpredictable way.
In order to indicate the correct branch, the 
expansions below are used.}
\vskip 2mm

\RU{Для построения начального приближения, я использую различные разложения функции ArcTra, в зависимости от значения аргумента $z$ в формуле \rf{traf}. Сейчас я их перечислю.}%
\EN{Expansions of tra in vicinity of the saddle points $\pm \pi \mathrm i$ can be inverted, giving the expansions of ArcTra in vicinity of 
$-1\pm \mathrm i \pi$. These expansions determine the positions of the branch points, and the direction of the cut lines, seen in figure
\ref{arctranmap}.}
\vskip 2mm

\EN{Consideration of the exponential in \rf{traagain} as a small parameter and as a big parameter gives two more asymptotic expansions.
Together with the Taylor expansion at unity and the expansions at the branch points, these expansions cover all the complex plane with the appropriate primary approximations, providing several correct decimal digits of the primary estimate $f_0$ of value of function ArcTra. Then, few iterations 
by \rf{arctranewton} already reach the maximal precision (15 
digits),
available for the complex double variables.
This algorithm is used in the numerical implementations of ArcTra and AuTra.
\footnote{
Details of the algorithm and the maps of the primary approximations for ArcTra  are loaded to the special page \url{https://mizugadro.mydns.jp/t/index.php/ArcTra} .}}
\vskip 3mm

\RU{Проще всего  арктраппманн раскладывается вблизи отрицательной части вещественной оси.
Такое разложение дает на первичную аппроксимацию}%
The easiest expansion is at the negative part of the real axis:
\be
\mathrm{ArcTra}(z) \approx 
\mathrm{app}_4(z)=z-\mathrm e^z
+\mathrm e^{2z}
-\frac{1}{2}\mathrm e^{3z}
\ee
The Readers are invited to plot the map of the agreement function 
\be
\mathcal A_4(z)= 
-\lg \left(
\frac{  |\mathrm{tra}(\mathrm{app_4}(z))-z| }
       { |\mathrm{tra}(\mathrm{app_4}(z))| + |z|}
\right)
\ee
\vskip 4mm

The logarithmic growth of function ArcTra can be caught with the asymptotic expansion
\JP{
\be\!\!\!\!\!\!
\mathrm{ArcTra}(z)= L\,\left( 1
+\frac{-1}{z}
+\frac{1\!-\!L/2}{z^2}
+\frac{-1+3 L/2 -L^2/3}{z^3}
+\frac{1-3L+11L^2/6-L^3/4}{z^4}
\right)+.. ~ ~
\ee
где
$L=\ln(z)$.
}%JP
This expansion provides the approximation
\be
\mathrm{ArcTra}(z)\approx  \mathrm{app_3}(z) =  
\ln(z) \left(1+ \frac{1}{z} \sum_{m=0}^M \frac{P_m(\ln(z))}{z^m}\right)
\ee
where $P_m$ is polynomial of $m$th order. In particular,
\be
P_0(\ell)&=&\!\! -1 \\
P_1(\ell)&=&~ 1-\ell/2 \\
P_2(\ell)&=&\!\!-1+3 \ell/2 -\ell^2/3  \\
P_3(\ell)&=&~ 1-3L+11\ell^2/6-\ell^3/4  \\
P_4(\ell)&=&\!\! -1+5\ell-35\ell^2/6+25\ell^3/12-\ell^4/5 
\ee
\vskip 4mm

The Taylor expansion of function tra at unity gives the approximation
\be
\mathrm{ArcTra}(z) \approx \mathrm{app}_1(z) =\sum_{n=1}^M c_n (z-1)^n
\eL{arctrap1}
Some tens of coefficients $c$ of this expansion can be calculated, inverting the Taylor expansion of function tea at zero; the beginning of the expansion can be written as follows:
\vskip 2mm

$\mathrm{ArcTra}(1\!+\!z)\!=\!\frac{z}{2}-\frac{z^2}{16}$ $
+\frac{z^3}{192}$ $
+\frac{z^4}{3072}$ $
-\frac{13 z^5}{61440}$ $
+\frac{47 z^6}{1474560}$ $
+\frac{73 z^7}{41287680}$ $
-\frac{2447 z^8}{1321205760}\!+..$\JP{ $

+\frac{16811 z^9}{47563407360} + .. $ $
+\frac{15551 z^{10}}{1902536294400}$ $
-\frac{1726511 z^{11}}{83711596953600}$ $
+\frac{18994849 z^{12}}{4018156653772800}$ $
-\frac{10979677 z^{13}}{208944145996185600}$ $
-\frac{2983409137 z^{14}}{11700872175786393600}$ $
+\frac{48421103257 z^{15}}{702052330547183616000}$ $
-\frac{135002366063 z^{16}}{44931349155019751424000}$ $
-\frac{778870772857 z^{17}}{235025518657026392064000}$ $
+\frac{232033147779359 z^{18}}{219983885462976702971904000}$ $
-\frac{1305952009204319 z^{19}}{16718775295186229425864704000}$ $
-\frac{58740282660173759 z^{20}}{1337502023614898354069176320000}$ $
+\frac{1862057132555380307 z^{21}}{112350169983651461741810810880000}$ $
+O\left(z^{22}\right)$
}%JP
\vskip 2mm
At $M\!=\!21$, for  $|z|\!<\!1$, this approximation provides at least 12 significant figures.
\vskip 4mm

Expansion of ArcTra at point $-1-\mathrm i \pi$ can be written as follows:
\be \!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
\mathrm{ArcTra}(z)\!
&\!\approx\!& \mathrm{app}_2(z)= 
\sum_{m=1}^{M} c_m^{~*} \, (z\!+\!1\!+\!\mathrm i \pi)^{m/2}
\nonumber \\
&\!=\!&\!
-\mathrm i \pi
+\mathrm i \sqrt{2} \sqrt{z\!+\!\mathrm i \pi\! +\!1}
+\frac{1}{3} (z\!+\!\mathrm i \pi \!+\!1)
\nonumber \\
&&\!
-\frac{\mathrm i (z\!+\!\mathrm i \pi \!+\!1)^{3/2}}{9 \sqrt{2}}
-\frac{2}{135} (z\!+\!\mathrm i \pi \!+\!1)^2
+.. 
\ee
\JP{$
+\frac{\mathrm i (z+\mathrm i \pi +1)^{5/2}}{540 \sqrt{2}}$ $
-\frac{4 (z+\mathrm i \pi +1)^3}{8505}$ $
+\frac{139 \mathrm i (z+\mathrm i \pi +1)^{7/2}}{340200 \sqrt{2}}$ $
+\frac{2 (z+\mathrm i \pi +1)^4}{25515}$ $
-\frac{571 \mathrm i (z+\mathrm i \pi +1)^{9/2}}{73483200 \sqrt{2}}$ $
+\frac{1124 (z+\mathrm i \pi +1)^5}{189448875}$ $
+O\left((z+\mathrm i \pi +1)^{11/2}\right)$ }%JP
In more compact (and more efficient for the numerical implementation) form, this expansion can be re-written as follows:
\be \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!
\mathrm{ArcTra}\big(\!-1 \!-\!\mathrm i \pi \!+\! 2 t^2\big)&=&
-\mathrm i \pi
+2 \mathrm i t
+\frac{2 t^2}{3}
-\frac{2 \mathrm i t^3}{9}
-\frac{8 t^4}{135}
\nonumber \\ &&
+\frac{\mathrm i t^5}{135}
-\frac{32 t^6}{8505}
+\frac{139 \mathrm i t^7}{42525}
+..
\ee
with 
obvious
modification for the ``conjugated'' region
\be \!\!\!\!\!\!\!\!\!\!
\mathrm{ArcTra}\Big(-1 + \mathrm i \pi + 2 t^2\Big)&=&
\mathrm i \pi
-2 \mathrm i t
+\frac{2 t^2}{3}
+\frac{2 \mathrm i t^3}{9}
-\frac{8 t^4}{135}
\nonumber \\ &&
-\frac{\mathrm i t^5}{135}
-\frac{32 t^6}{8505}
-\frac{139 \mathrm i t^7}{42525}
+..
\ee
where $t$ has sense of algebraic function of argument $z$ of the arctrappmann,
\be
t=\sqrt{\frac{z+1-\mathrm i \pi}{2}}
\eL{arctrait}
The cut of the square root function in \rf{arctrait} automatically determines the cut lines of function ArcTra, 
seen in figure
\ref{arctranmap}.
I invite the Reader to plot the maps of the agreement functions
\be
\mathcal A_m(z)= \displaystyle 
-\lg \left(
\frac{  |\mathrm{tra}(\mathrm{app}_m(z))-z| }
       { |\mathrm{tra}(\mathrm{app}_m(z))| + |z|}
\right)
\ee
in the complex plane
$z\!=\!x\!+\!\mathrm i y$ 
for
$m=1,2,3,4$, 
id est, for the four primary approximations suggested in this section.
\vskip 2mm

When the approximations above are implementd and called with names
arctra1,
arctra2,
arctra3,
arctra4,
the numeral implementation of function ArcTra can follow the algorithm below:
\begin{verbatim}
z_type arctran(z_type z) { DB x=Re(z), y=Im(z);
if( x>2.) return arctra3(z);
DB Y=fabs(y);
if(Y<M_PI){ if(x<-1.5) return arctra4(z);
            if(Y<2.) return arctra1(z); }
if( Y>5. || x<-4. ) return arctra3(z);
if( y>0. ) return arctra2(z);
return conj( arctra2(conj(z)) );
}
\end{verbatim}

Alternatively, function
ArcTra can be expressed through the Tania function \rf{taniaprim}:
\be
\mathrm{ArcTra}(z)=z-\mathrm{Tania}(z\!-\!1)
\eL{ArcTraTania}
In such a way, in this Book, function Tania is used already 3 times in three pretty different ways:
First, in the simple model of the laser amplifier,
then, in the representation of function flog, and, en fin, now, as an alternative implementation of ArcTra.
\vskip 6pt

At least on vicinity of the real axis, function Tania can be expressed also through other special functions namely, 
LambertW 
and
WrightOmega, 
see 
\rf{TaniaWright}.
In principle, one could use those representations instead of the implementation described above.
However, at large $z$, Tania shows the growth similar to the proportional; so, the numerical implementation of ArcTra through Tania may cause loss of precision due to the rounding errors.
However, the special implementation of ArcTra specifid above, had been used for the testing of the numerical implementation of function Tania.
\vskip 6pt


While both tra and ArcTra are described and implemented,
function tra can be iterated, using the representation through the superfunction and the abelfunction.
I call these functions SuTra and AuTra; then, as usually,
\be
\mathrm{tra}^n(z)=\mathrm{SuTra}\Big(n+\mathrm{AuTra}(z)\Big)
\eL{tranz}
In such a way I announce the future consideration:
Function SuTra is 
described
in the next section, and function AuTra is treated soon after that section.

\section{Supertrappmann}
\baselineskip17.8pt

\begin{wrapfigure}{r}{250px}
\vskip -28pt
\sx{.315}{\begin{picture}(602,800)
\normalsize
\put(510,794){\sx{4}{$y$}}
\put(420,566){\sx{3.8}{$1\!+\!\mathrm e$}}
\put(510,690){\sx{4}{$5$}}
\put(510,590){\sx{4}{$4$}}
\put(510,490){\sx{4}{$3$}}
\put(510,390){\sx{4}{$2$}}
\put(510,289){\sx{4}{$1$}}
\put(512,187){\sx{4}{$0$}}
\put(490, 87){\sx{4}{$-\!1$}}
\put(068,206){\sx{4}{$-4$}}
\put(168,206){\sx{4}{$-3$}}
\put(200,180){\sx{4}{$-\mathrm e$}}
\put(268,206){\sx{4}{$-2$}}
\put(368,206){\sx{4}{$-1$}}
\put(491,206){\sx{4}{$0$}}
\put(592,206){\sx{4}{$1$}}
\put(692,206){\sx{4}{$2$}}
\put(778,206){\sx{4}{$x$}}
\put(662,308){\sx{4.2}{\rot{70}$y\!=\!\mathrm{SuTra}(x)$\ero}}
\put(240,318){\sx{3.8}{\rot{0}$y\!=\!-\!\ln(-x)$\ero}}
\put(0,0){\ing{sutraplo}}
\end{picture}}
\vskip -4pt
\sx{.61}{\url{https://mizugadro.mydns.jp/t/index.php/File:SuTraPlo3T.jpg}}
\vskip -6pt
\caption{$y\!=\!\mathrm{SuTra}(x)$ and $y\!=\!-\ln(-x)$  \label{sutraplot}}
\vskip -2pt
\end{wrapfigure}

As it is mentioned above, the Trappmann function tra has no fixed points.
For this reason, the methods described in the previous chapters of this Book, cannot be applied ``as is''.
Henryk Trappmann had expected, that it will be difficult, to construct superfunction for the function tra, if at all.
\vskip 6pt

However, the simple and efficient way to construct and evaluate SuTra, which is superfunction for the Trappmann 
function, exist. Plot of function SuTra is shown in figure \ref{sutraplot} and compared to the graphic of its asymptotic.
Below I describe the construction of this function.
\vskip 6pt
 
Many colleagues, instead of to trace the deduction, ask first ``How did you guess?'' Following such an interest, 
first, I describe the hint, that leads to the efficient representation of function SuTra.
For the transfer function tra, and its superfunction $F$, the transfer equation can be written as follows:
\be\!\!\!\!\!\!\!\!\!
F(z\!+\!1)=\mathrm{tra}(F(z))=F(z)+\exp(F(z))
\eL{trantrap}
and re-written in the following form:
\be
F(z\!+\!1)-F(z)=\exp(F(z))%\approx 1- F(z)
\eL{trantrap1}
In the left hand side of equation  \rf{trantrap1}, I see something, that looks similar to derivative of function $F$.
This similarity can be expressed with approximate equation
\be
\frac{\mathrm d F}{\mathrm d z} \approx \frac{1}{\exp(-F)}
\ee
that gives
\be
\int \exp(-F) \mathrm d F \approx \int \mathrm d z
\ee
\be
\exp(-F) \approx - z
\ee
This relation indicates, that expansion of superfunction $F$ may begin with the logarithmic term, namely the term,
shown in figure \ref{sutraplot}.

However, the ``heuristic solution'' above is not a solution at all; in the best case, it is only the asymptotic approximation.
Substitution of the approximation of function $F$ into the transfer equation
\rf{trantrap} gives, of course, some residual. 
This residual indicates the form of the next term of the expansion; and so on.
In such a way, I guess the form of the asymptotic solution $F$:
\be
F(z)=F_M(z)+O\left( \frac{\ln(-z)}{z}\right)^{M+1}
\eL{sutraFMO}
where $M$ is natural number, and
\be \!\!\!\!\!\!\!\!
F_M(z)=-\ln(-z)+\sum_{m=1}^{M} P_m\Big(\ln(-z)\Big) \, z^{-m}
\eL{sutraFM}
where 
\be
P_m(z)=\sum_{n=1}^m a_{m,n} z^n
\eL{sutraP}
Substituting this expansion into the transfer equation \rf{trantrap}, I collect terms with equal powers of $z$ and equal powers of  $\ln(-z)$. This gives both, verification of the form of the asymptotic expansion \rf{sutraFM},\rf{sutraP}
and values of coefficients $a$.
Several coefficients $a$, calculated in this way,
are shown in Table 
\ref{table:sutracoe}.
\begin{table}[h]
\caption{\EN{Coefficients $a$ in the expansion}\RU{Коэффициенты $a$ в разложении} \rf{sutraFMO},\rf{sutraFM} \label{table:sutracoe}}
\sx{1.14}
{$
\begin{array}{ccccccccccc}
 0 & -\frac{1}{2} & a_{1,2} & a_{1,3} & a_{1,4} & a_{1,5} & a_{1,6} & a_{1,7} & a_{1,8}  \rule{0pt}{14pt} \\
 \frac{1}{6} & -\frac{1}{4} & \frac{1}{8} & a_{2,3} &  a_{2,4} & a_{2,5} & a_{2,6} & a_{2,7} & a_{2,8} \rule{0pt}{14pt} \\
 \frac{7}{48} & -\frac{7}{24} & \frac{3}{16} & -\frac{1}{24} & a_{3,4} & a_{3,5} & a_{3,6} & a_{3,7} &  a_{3,8} \rule{0pt}{14pt} \\
 \frac{647}{4320} & -\frac{35}{96} & \frac{5}{16} & -\frac{11}{96} & \frac{1}{64} & a_{4,5} & a_{4,6} & a_{4,7} & a_{4,8} \rule{0pt}{14pt}  \\
 \frac{1427}{8640} & -\frac{4163}{8640} & \frac{25}{48} & -\frac{17}{64} & \frac{25}{384} & -\frac{1}{160} & a_{5,6} & a_{5,7} & a_{5,8}  \rule{0pt}{14pt} \\
 \frac{1380863}{7257600} & -\frac{1883}{2880} &  \frac{5963}{6912} & -\frac{653}{1152} & \frac{305}{1536} & -\frac{137}{3840} & \frac{1}{384} & a_{6,7} & a_{6,8}  \rule{0pt}{14pt} \\
 \frac{3278773}{14515200} & -\frac{2171723}{2419200} & \frac{97603}{69120} & -\frac{3961}{3456} & \frac{537}{1024} & -\frac{263}{1920} & \frac{49}{2560}  & -\frac{1}{896} & a_{7,8}  \rule{0pt}{14pt} \\
 \frac{251790467}{914457600} & -\frac{35981749}{29030400} & \frac{1049251}{460800} & -\frac{920881}{414720} & \frac{69953}{55296} & -\frac{13381}{30720} & \frac{4123}{46080} & -\frac{363}{35840} & \frac{1}{2048}  \rule{0pt}{14pt} 
\end{array}$}
\end{table}

\vskip 3mm

Using Mathematica, coefficients $a$ can be found with the following code:
\normalsize
{\baselineskip 15pt
\begin{verbatim}
T[z_] = z + Exp[z];
Clear [n, m, M];
P[m_, L_] := Sum[a[m, n] L^n, {n, 0, m}]; P[m, L];
F[z_]=-Log[-z]+a[1,1] Log[-z]/z+Sum[P[m,Log[-z]]/z^m,{m,2,M}]
M = 12;
F1x = F[-1/x + 1];
Ftx = T[F[-1/x]];
s[1] = Series[F1x - Ftx, {x, 0, 2}];
t[1] = Extract[Solve [Coefficient[s[1], x^2] == 0, {a[1, 1]}], 1]
A[1, 1] = ReplaceAll[a[1, 1], t[1]];
su[1] = t[1]

m = 2; s[m] = ReplaceAll[Series[F1x - Ftx, {x, 0, m + 1}], su[m]];
t[m] = Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m + 1)];
u[m] = Collect[t[m], L];
v[m] = Table[Coefficient[u[m] L, L^(n + 1)] == 0, {n, 0, m}];
w[m] = Table[a[m, n], {n, 0, m}];
ad[m] = Extract[Solve[v[m], w[m]], 1];
su[m + 1] = Join[su[m], ad[m]];
ReplaceAll[ReplaceAll[F[x], su[m + 1]], Log[-x] -> L]

m = 3; s[m] = ReplaceAll[Series[F1x - Ftx, {x, 0, m + 1}], su[m]];
t[m] = Coefficient[ReplaceAll[s[m], Log[x] -> L], x^(m + 1)];
u[m] = Collect[t[m], L];
v[m] = Table[Coefficient[u[m] L, L^(n + 1)] == 0, {n, 0, m}];
w[m] = Table[a[m, n], {n, 0, m}];
ad[m] = Extract[Solve[v[m], w[m]], 1];
su[m + 1] = Join[su[m], ad[m]];
ReplaceAll[ReplaceAll[F[x], su[m + 1]], Log[-x] -> L]
\end{verbatim}
}
\large
\baselineskip 19.8pt
\vskip 2mm
and so on
$m\!=\!4$, $m\!=\!5$, etc.
I do not arrange here the loop ``For'' with respect to $m$, in order to keep the code explicit and simplify the tracing step by step.
\vskip 2mm

Expression \rf{sutraFM} can be considered as primary approximation of superfunction of the Trappmann function \rf{tra}. Then, the exact
solution $F$ of the transfer equation 
appears as limit
\be \!\!\!\!\!\!\!\!
F(z)=\lim_{k \rightarrow \infty} \mathrm{tra}^k\big(F_M(z\!-\!k)\big)
\eL{exa}
In order to get superfunction SuTra, that satisfies also the additional condition
\be
\mathrm{SuTra}(0)=0
\eL{SuTra0}
I set
\be
\mathrm{SuTra}(z)=F(z+x_0)
\eL{SuTraDef}
where $x_0 \!\approx\! -1.1259817765745026$ is real solution of equation
\be
F(x_0)=0
\eL{x00}
Figure \ref{sutraplot} shows $y=\mathrm{SuTra}(x)$ versus $x$.
For comparison, the thin curve shows the leafing term of the asymptotic representation of  $\mathrm{SuTra}$, 
id est, 
$y=-\ln(-x)$. 
In the left hand side of the figure, graphic of SuTra approaches its asymptotic.
\begin{figure}[t]
\normalsize
\hskip 21pt
\sx{.2}{\begin{picture}(2002,2002)
\put(-49,1950){\sx{8}{$y$}}
\put(-49,1780){\sx{7}{$8$}}
\put(-49,1580){\sx{7}{$6$}}
\put(-49,1380){\sx{7}{$4$}}
\put(-49,1180){\sx{7}{$2$}}
\put(-49,0980){\sx{7}{$0$}}
\put(-109,780){\sx{7}{$-2$}}
\put(-109,580){\sx{7}{$-4$}}
\put(-109,380){\sx{7}{$-6$}}
\put(-109,180){\sx{7}{$-8$}}
\put(120, 6){\sx{7}{$-8$}}
\put(320, 6){\sx{7}{$-6$}}
\put(520, 6){\sx{7}{$-4$}}
\put(720, 6){\sx{7}{$-2$}}
\put(988, 6){\sx{7}{$0$}}
\put(1188, 6){\sx{7}{$2$}}
\put(1388, 6){\sx{7}{$4$}}
\put(1588, 6){\sx{7}{$6$}}
\put(1788, 6){\sx{7}{$8$}}
\put(1958, 6){\sx{8}{$x$}}
\put(0,0){\ing{sutranmap}}
\put(200,1414){\sx{7}{\rot{56}$u\!=\!-2.2$\ero}}
\put(352,1328){\sx{7}{\rot{56}$u\!=\!-2$\ero}}
\put(462,1238){\sx{7}{\rot{56}$u\!=\!-1.8$\ero}}
\put(574,1188){\sx{7}{\rot{56}$u\!=\!-1.6$\ero}}
\put(652,1130){\sx{7}{\rot{56}$u\!=\!-1.4$\ero}}
\put(722,1070){\sx{7}{\rot{58}$u\!=\!-1.2$\ero}}
\put(790,1050){\sx{7}{\rot{59}$u\!=\!-1$\ero}}
\put(1518,1200){\sx{7}{\rot{10}$v\!=\!3$\ero}}
\put(1277,1590){\sx{7}{\rot{68}$v\!=\!2$\ero}}
\put(618,1632){\sx{7}{\rot{-53}$v\!=\!1$\ero}}
\put(340,1128){\sx{7}{\rot{-10}$v\!=\!0.2$\ero}}
\put(310,0984){\sx{7}{$v\!=\!0$}}
\put(330,0832){\sx{7}{\rot{10}$v\!=\!-0.2$\ero}}
\put(642,0332){\sx{7}{\rot{53}$v\!=\!-1$\ero}}
\put(1209,488){\sx{7}{\rot{-70}$v\!=\!-2$\ero}}
\put(1490,772){\sx{7}{\rot{-11}$v\!=\!-3$\ero}}
\end{picture}}
\vskip -1pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:SuTraMapT.jpg}
\vskip -6pt
\caption{$u\!+\!\mathrm i v=\mathrm{SuTra}(x\!+\!\mathrm i y)$ по формуле \rf{SuTraDef} \iL{sutramap}}
\end{figure}
\vskip 2mm

Complex map of function $\mathrm{SuTra}$ is shown in figure \ref{sutramap}. 
Being far from the positive part of the real axis, function SuTra looks similar to
function $z \rightarrow -\ln(-z)$, 
as it is suggested by the leading term of its asymptotic representation.
Lines of the constant real part look similar to circles, while lines of constant imaginary part look similar to the straight lines;
this make the map to look similar to the map of logarithm. 
\vskip 2mm

\vskip 2mm
\RU{
В правой части рисунка \ref{sutramap}, в окрестности положительной части вещественной оси, функция $\mathrm{SuTra}$ остается голоморфной, но она очень быстро растет (быстрее любой экспоненты), и даже самый хороший графопостроитель не может провести соответствующие линии уровня. (И даже если бы он смог такое сделать, получающийся файл не поместился бы ни в один компьютер.) Поэтому эта часть рисунка осталась пустой, эта пустота выглядит как ручка, на которой держится голоморфная иммитация сингулярности в нуле, и, в некотором, смысле, вся ТФКП. Потому как если кто построит целую голоморфную функцию с логарифмической асимптотикой, но без такой ``ручки'', то вся  ТФКП (с аксиомами Арифметики вместе) посыпется. Такая ``ручка'' неизбежна при аппроксимации сингулярной функции целой функцией.}
\vskip 2mm

\begin{figure}[t]
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\end{picture}}
\vskip -2pt
{
\url{https://mizugadro.mydns.jp/t/index.php/File:Sutralomap.jpg}}
\vskip -8pt
\caption{\large
Complex maps of functions $f_2$ by
\rf{sutra2} 
and $f_4$ \rf{sutra4}, 
overlapped with map of function $f_\infty$ by
\rf{sutrafinfty}
\label{sutralomap}}
\vskip 8pt
\end{figure}
I would like to show, how the logarithmic function
$z\rightarrow -\ln(-z)$ 
can be approximated with the entire function.
Figure \ref{sutralomap} shows complex maps of the entire functions
\be
f_2(z)=\mathrm {SuTra}(2z)+\ln(2) \label{sutra2}\\
f_4(z)=\mathrm {SuTra}(4z)+\ln(4) \label{sutra4}
\ee
for comparison, each of these maps is overlapped with the map of 
\be
f_{\infty}(z)=-\ln(-z) \label{sutrafinfty}
\ee
In the right hand side picture of figure  \rf{sutralomap}, the levels of different functions are so close, that it is difficult to see,
which of functions does each level corresponds to.
For all 
$z$, 
except zero and positive part of the real axis, sequence 
\be
\Phi_n(z)=-\Big(\mathrm {SuTra}(-nz)+\ln(n) \Big) \label{sutn}
\eL{phin}
at big $n$ 
approximates
$\ln(z)$.
\RU{
\footnote{В окрестности положительной части вещественной оси, при конечном $n$,  функция $\Phi_n$ сильно дергается; по сравнению с ней, даже выживший из ума 
диктатор, оказывающий всем соседям ``братскую помощь'', выглядит образцом постоянства и здравомыслия.} 
}%
Up to my knowledge, expression \rf{sutn} provides the range of approximation of logarithm with entire functions,
that is wider than that of all approximations ever suggested before publication 
\cite{sutra}.
\vskip 2mm

I have no idea, what for the approximation of logarithm with entire function can be used.
This approximation appears here as a by-product at construction of function SuTra.
But if someone needs such an approximation, it is done and it is here,
formula \rf{phin}.
\section{Implementation of SuTra and the testing}
\baselineskip 21pt

The key parameters of any approximation are the range of applicability, the precision and the
number of elementary operations that should be performed for each evaluation. 
If the range of applicability is wide (for example, cover the range of holomorphism of the approximated function), the precision is close to the that allowed at the computer representation of numbers, and
the complex maps can be plotted in real time, then I qualify the solution as ``exact''.
This implies, that, if, for some reason, the precision needs to be improved, the implementation of the high precision code is straightforward, id est, can be realised with the same formulas.

\vskip 2mm

If some function is supplied with definition, the properties are described, and the efficient precise algorithm for the evaluation is presented, I treat such a function as a ``special function''.
Then, if the solution of some problem is explicitly expressed in terms of the special functions,
I call it ``exact solution''.
I place this 
explanation here form my old coauthor, who believed, that even $\pi$ is approximate number.
However, the fundamental mathematical constants are, contrary, exact numbers, in the sense mentioned above; neither $\pi$, nor values of superfunctions constrcted in this Book are exceptions:
they are exact and can be evaluated with any required precision.
\vskip 2mm

Following the ideology of the preamble of this section, I consider here
the range of values of the argument $z$, at which the approximation $F_M(z\!+\!x_0)$
by \rf{sutraFM} 
can be considered as good approximation of
$\mathrm{SuTra}(z)$. 
Then I describe the algorithm, based on this analysis.
\vskip 2mm

For the practical reasons (in order to get the complex double implementation, that can be easily reproduced and verified), I took 12 terms in the expansion, id est, $M\!=\!12$.
For the dozen terms, the agreement
\be
\mathcal A(z)= - \lg\! \left( \frac
{ |  F_M(z\!+\!x_0)- \mathrm {SuTra}(z) | }
{ |  F_M(z\!+\!x_0) | + |\mathrm {SuTra}(z) | }
\right)
\eL{sutraA}
had been analysed. Levels of this function are shown in figure 
\ref{sutraAmap}.
\vskip 2mm

\begin{figure}[t]
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\put(2030,0488){\sx{16}{$\mathcal A \!=\!9$}}
\end{picture}}
\vskip 8pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Sutraamap.jpg}
\vskip -2pt
\caption{
Map of $\mathcal A=\mathcal A(x\!+\!\mathrm i y)$  by \rf{sutraA} \label{sutraAmap}}
\vskip 6pt
\end{figure}

For the precision complex double, %id est with approximately 15 decimal digits, the
values outside the ``thick'' contour in figure \ref{sutraAmap} can be used ``as is''.
For other values, formula 

\be
\mathrm{SuTra} (z)=\mathrm{tra}^k(\mathrm{SuTra}(z\!-\!k))
\eL{SuTrak}
is used with appropriate 
$k$,
in such a way, that the inner argument is outside the ``thick'' contour in figure
\ref{sutraAmap}. 
This thick contour is not part of the map; it is formed with the segment along the line
$x\!=\!-11$,
arc with centre at point $(5,0)$ of radius 18, and the half-line along $y\!=\!6$.

Outside the thick contour in figure \ref{sutraAmap}, the 
precision
of evaluation of function SuTra 
is limited
mainly by the rounding errors. 
Values outside the ``thick contour'' are used for the numerical implementation directly. For other values, 
formula \ref{SuTrak} is used with appropriate value of $k$. This algorithm is used for the numerical implementation of function SuTra,
\url{https://mizugadro.mydns.jp/t/index.php/Sutran.cin}
\vskip 6pt

At the evaluation of function SuTra,
function tra should be evaluated of order of ten times. (However, this depend on the initial value).
Function tra is fast, because $\mathrm{tra}(z)\!=\!z\!+\!\exp(z)$.
In such a way, the evaluation of SuTra is only for an order of magnitude slower, than evaluation of other special functions like exp, erfc or BesselJ.
This is one of reasons why I qualify SuTra as special function.
\vskip 6pt

For iterates of the Trappmann function, I need also the inverse function, $\mathrm{AuTra}={\mathrm{SuTra}^{}-1}$. This function is shown in figure \ref{autraplot}
and described in the next section.

\newpage
\section{AuTra, abelfunction of trappmann}
\baselineskip 21pt

\begin{figure}[t]
\normalsize
\vskip -2pt
\hskip -4pt
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\put(112,296){\sx{2.2}{$1$}}
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\put(0,0){\ing{autraplo}}
\put(154,304){\sx{2}{\rot{35}$y\!=\!c_0\!+\!x/2\!-\!\mathrm e^{-x}$\ero}}
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\put(170,122){\sx{2}{$y\!=\!c_0\!+\!x/2\!-\!\mathrm e^{-x}\!-\!\mathrm e^x/6$}}
\end{picture}}
\vskip -6pt
{\url{https://mizugadro.mydns.jp/t/index.php/File:Autraplot.jpg}}
\vskip -8pt
\caption{
$y\!=\!\mathrm{AuTra}(x)$ 
by \rf{AuTraDef}
and two %~ ~ ~ ~ ~ ~  
its asymptotics by 
\rf{sutrasy} 
\label{autraplot}}
\vskip 8pt
\end{figure}
This section describes evaluation of the inverse function of SuTra,
id est, the Abel function for the Trappmann function tra.
I call this abelfunction AuTra. Its explicit plot is shown in figure
\ref{autraplot}; for comparison, in the same picture, the two asymptotics of function AuTra are shown.
\vskip 2mm

Complex map of function AuTra is shown in figure \ref{autraMap}. 
\vskip 2mm

For the transfer function tra, the abelfunction $G$ satisfies the Abel equation
\be
G(\mathrm{tra}(z))=G(z)+1
\eL{abeltraeq}
In order to see the asymptotic expansion of the solution $G$, I rewrite this equation as
\be
G(z+\mathrm e^z)-G(z)=1
\eL{abeltraeq1}
and expand the left hand side, using $\mathrm e^z$ as small parameter: 
\be
G'(z) \mathrm e^z +G''(z) e^{2z}/2+.. \approx 1
\eL{abeltraeq3}
This expansion allows to guess the asymptotic solution:
\be \!\!\!\!\!\!\!\!\!\!
G(z)\approx \frac{z}{2}-\mathrm e^{-z}
-\frac{\mathrm e^z}{6}
+\frac{\mathrm e^{2 z}}{16}
-\frac{19 \mathrm e^{3 z}}{540}
+\frac{\mathrm e^{4 z}}{48}
-\frac{41 \mathrm  e^{5 z}}{4200}
+ \mathcal O( \mathrm  e^{6 z})
\label{sutrasy}
\ee
\EN{The coefficients of the expansion above  are calculated and evaluated with the Mathematica code below:}%
\RU{Коэффициенты разложения \rf{sutrasy} получены с помощью Математики, 
использована вот такая программа:}%
\normalsize 

\baselineskip 15pt
\begin{verbatim}
tra[z_]=z+Exp[z];
g0[z_] = z/2 - Exp[-z] + Sum[c[n] Exp[n z], {n, 1, 20}]
n = 1; s[n] = Series[g0[Log[t]] + 1 - g0[tra[Log[t]]], {t,0,n+1}]
u[n] = Extract[Solve[Coefficient[s[n], t^(n+1)] == 0, c[n]], 1]
g[n, z_] = ReplaceAll[g0[z], u[n]]
For[n = 1, n < 20, n++;
   s[n] = Series[ g[n-1,Log[t]]+1-g[n-1,tra[Log[t]]],{t,0,n+1}];
   u[n] = Extract[Solve[Coefficient[s[n], t^(n+1)] == 0, c[n]],1];
   g[n,z_] = ReplaceAll[g[n-1, z], u[n]];        ]
g[n, z]
Table[Coefficient[g[n, z], Exp[n z]], {n, 1, 20}]
N[Table[Coefficient[g[n, z], Exp[n z]], {n, 1, 20}], 18]
\end{verbatim}
\large

\baselineskip 18pt
The same coefficients can be obtained also by the inversion of the asymptotic expansion of function SuTra.
Note, that the asymptotic expansion of AuTra is simpler, than that of SuTra.
\vskip 2mm

\EN{For some fixed integer $M$, define the primary approximation as truncation of the series above:}%
\RU{Для некоторого натурального числа $M$, я определяю функцию $G_M$ как оборванную сумму для разложения \rf{sutrasy}; пусть}%
\be
G_M(z)=\frac{z}{2}-\mathrm e^{-z}
\displaystyle +\sum_{m=1}^{M} c_m \mathrm e^{mz} 
\eL{autraGM}
\EN{Define function $G$ as limit}%
\RU{В этой секции, я определяю функцию $G$ как предел}%
\be
G(z)=\lim_{n\rightarrow \infty} (G_M(\mathrm{ArcTra}^n(z))+n)
\eL{autraG}
AuTra can be expressed through $G$ with
\be\!\!\!\!
\mathrm{AuTra}(z)=G(z)-G(0) \approx G(z)+1.1259817765745026
\label{AuTraDef}
\ee
The constant $G(0)$ can be interpreted also as negative of coefficient $x_0$ in equation \rf{x00}, id est, $x_0\!=\!-G(0)$.
\vskip 2mm

For the numerical implementation, we need to choose the appropriate number $M$ in 
and to determine the number $n$ of iterations approximating limit in equation \rf{autraG}. 
For the complex double arithmetics, the reasonable choice is $M\!=\!9$. Then, the primary approximation 
\be
\mathrm{SuTra}(x+\mathrm i y)\approx g_9(x+\mathrm i y)
\label{SuTrappro}
\ee
is used for the region defined with condition
\be
|y|\!<\!3 ~ ~ \mathrm{and} ~ ~ |y|/3\!+\!x\!<\! 3.5
\label{sutraregion}
\ee

\begin{figure}[t]
\normalsize
\begin{center}
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\put(386,360){\sx{3}{\rot{90}$\mathcal A\!=\!9$\ero}}
\put(440,360){\sx{3}{\rot{90}$\mathcal A\!=\!6$\ero}}
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\put(542,260){\sx{3}{\rot{90}$\mathcal A\!=\!1$\ero}}
\put(528,390){\sx{2.8}{$\mathcal A \!<\! 1$}}
\put(166,142){\sx{2.8}{$\mathcal A \!\approx\! 16$}}
\end{picture}}
\end{center}
\vskip -12pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:Autran0m9tes64t.jpg}
\vskip -6pt
\caption{\large
Map of agreement  $\mathcal A=A(x\!+\!\mathrm i y)$  by  \rf{AuTraAgre}; range \rf{sutraregion} is shaded
$\mathcal A=A(x\!+\!\mathrm i y)$  по формуле  \rf{AuTraAgre}
\iL{autraAgreMap}}
\end{figure}

This region is shaded in the figure 
\ref{autraAgreMap} 
Also, at the same figure, the map of agreement $A$ is shown,

\be\!
\mathcal A(z)\!=\!-\lg\!\!\left( \frac{ |\mathrm{SuTra}(g_M(z))\!-\!z|}{ |\mathrm{SuTra}(g_M(z))|\!+\!|z|} \right)
\label{AuTraAgre}
\ee
Map in figure \rf{autraAgreMap} can be considered also as the numerical verification of relation 
\be \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\mathrm{SuTra}(\mathrm{AuTra}(z))=z 
\eL{SuTraAu}
In the shaded range, for the numerical implementation, the relation \rf{SuTraAu} holds with at least 15 decimal digits; as it is supposed to be while
AuTra is abelfunction, corresponding to superfunction SuTra for the transfer function tra.
\vskip 2mm

Region, where the primary approximation is used, can be optimised,
approaching to the level
$\mathcal A\!=\!15$
and improving the algorithm. I suggest, the Reader can do the as an exercise;
I hope, the reader will not forget to test the improved algorithm.
\vskip 2mm

The primary approximation \rf{SuTrappro} can be used only in the narrow range of values of the argument shaded in figure {autraAgreMap}.
If the initial argument $z\!=\! x\!+\!\mathrm i y$ happens to be outside the shaded region, then, the function $\mathrm{ArcTra}=\mathrm{Tra}^{-1}$ is applied $n$ times with such $n$, that the argument comes to the shaded range. Then, approximation
\be
\mathrm{AuTra}(z) \approx g_9( \mathrm{ArcTra}^n(z))+n  ~ ~ ~ ~
\eL{AuTrag9}
is used to evaluate the function. 
\vskip 2mm

The cuts of the range of holomorphism of function ArcTra determine also the cuts the range of holomorphism of function AuTra. 
These cuts are seen in figures \ref{arctranmap} 
and
\ref{autraMap}.
\RU{При больших значениях $|z|$, функция $\mathrm{AuTra}(z)$ медленно растет, за исключением полу-полосы $\Re(z)\!<\!0$, $|\Im(z)|\!<\!\pi$. В этой полуполосе, при больших отрицательных значениях $\Re(z)$, 
экспоненциально. В частности, это относится к поведению функции при вещественных значениях аргумента. Такое поведение согласуется со свойствами функции SuTra, график которой показан на рисунке \ref{sutraplot}.}%
\EN{At large $|z|$, function $\mathrm{AuTra}(z)$ shows the slow growth, except the half-strip $\Re(z)\!<\!0$, $|\Im(z)|\!<\!\pi$.
In this half-strip, at large negative values of $\Re(z)$, function  $|\mathrm{AuTra}(z)|$ increases exponentially.
In particular, this refers to the real values of the argument.
This behaviour agrees with that of SuTra shown in figure
\ref{sutraplot}.}%
\vskip 2mm

\begin{figure}[t]
\vskip -4pt
\begin{center}
\sx{.2}{\begin{picture}(2102,2094)
\put(98,88){\ing{autranma}}
\put(40,2048){\sx{8}{$y$}}
\put(40,1870){\sx{7}{$8$}}
\put(40,1670){\sx{7}{$6$}}
\put(40,1470){\sx{7}{$4$}}
\put(40,1270){\sx{7}{$2$}}
\put(40,1070){\sx{7}{$0$}}
\put(-12,870){\sx{7}{$-2$}}
\put(-12,670){\sx{7}{$-4$}}
\put(-12,470){\sx{7}{$-6$}}
\put(-12,270){\sx{7}{$-8$}}
\put(002, 28){\sx{7}{$-\!10$}}
\put(220, 28){\sx{7}{$-8$}}
\put(420, 28){\sx{7}{$-6$}}
\put(620, 28){\sx{7}{$-4$}}
\put(820, 28){\sx{7}{$-2$}}
\put(1088, 28){\sx{7}{$0$}}
\put(1288, 28){\sx{7}{$2$}}
\put(1488, 28){\sx{7}{$4$}}
\put(1688, 28){\sx{7}{$6$}}
\put(1888, 28){\sx{7}{$8$}}
\put(2058, 28){\sx{7}{$x$}}
\put(218,1384){\sx{10}{\bf cut}}
\put(218, 752){\sx{10}{\bf cut}}
\put( 224,1836){\sx{8}{\rot{35}$v\!=\!0.6$\ero}}
\put( 1590,1496){\sx{8}{\rot{70}$v\!=\!0.4$\ero}}
\put( 1800,1360){\sx{8}{\rot{38}$v\!=\!0.2$\ero}}
\put( 140,1068){\sx{9}{$v\!=\!0$}}
\put(1874,1068){\sx{9}{$v\!=\!0$}}
\put( 1740, 814){\sx{8}{\rot{-40}$v\!=\!-0.2$\ero}}
\put( 194, 306){\sx{8}{\rot{-35}$v\!=\!-0.6$\ero}}
\put( 606, 1920){\sx{9}{\rot{-64}$u\!=\!3$\ero}}
\put( 1500, 992){\sx{8.2}{\rot{90}$u\!=\!2$\ero}}
\put( 1626, 960){\sx{8}{\rot{90}$u\!=\!2.2$\ero}}
\put( 1842, 960){\sx{8}{\rot{90}$u\!=\!2.4$\ero}}
\put( 1724, 400){\sx{8}{\rot{22}$u\!=\!2.6$\ero}}
\put( 670, 290){\sx{9}{\rot{60}$u\!=\!3$\ero}}
\end{picture}}
\end{center}
\vskip -16pt
\hskip 24pt
\url{https://mizugadro.mydns.jp/t/index.php/File:AuTraMapT.jpg}
\vskip -6pt
\caption{\large $u\!+\!\mathrm i v=\mathrm{AuTra}(x\!+\mathrm i y)$
\iL{autraMap}}
\vskip -4pt
\end{figure}

\RU{В этой секции, в порядке численной проверки имплементации функции AuTra, используется функция функция SuTra.
Читaтель приглашается стащить имплементации этих функций и оценить невязки при подстановке таких приближeний в уравнение Абеля \rf{abeltraeq}  и передаточное уравнение \rf{trantrap}. Мне кажется, что я сделал эти имплементации близко к той точности, какую только можно получить для переменных complex double. }%
\EN{Readers are invited to download the generators of the figures of this chapter, together with implementations of SuTra and AuTra, 
and investigate numerically the ranges of applicability of identities
\be
\rm{AuTra}(\mathrm{SuTra}(z))=z ~, ~
\rm{SuTra}(\mathrm{AuTra}(z))=z
\ee
 }%
\vskip 2mm

\RU{С функциями SuTra и AuTra, потроенными в этой главе, нетрудно считать нецелые итерации функции Траппманна. Этому посвящена следующая секция.}%
\EN{With functions SuTra and AuTra, described in this chapter, one can evaluate the non-integer iterates of function tra.
These iterates are described in the next section.}%

\section{Iterates of trappmann}
\baselineskip 19pt

\RU{В этой секции
итерируется не сам Генрик Траппманн, а функция Траппманна $\mathrm{tra}(z)\!=\!z\!+\!\mathrm e^z$.
Как и для других передаточных функций, итерации выражаются через суперфункцию и абельфункцию; то есть, в данном случае, через SuTra и  AuTra, построенные в этой главе:}%
With functions SuTra and AuTra, the iterate of the Trappmann function can be expressed as follows:
\be
\mathrm{tra}^n(z)=\mathrm{SuTra}\Big(n+\mathrm{AuTra}(z)\Big)
\eL{traite}
As usually, the number $n$ of iterate has no need to be integer. For real value of the argument and some real values of the number $n$, the iterates of the Trappmann function are shown in figure 
\ref{traiteplot}.
\vskip 2mm

Iterates of function $\mathrm{tra}(z)=z\!+\!\mathrm e^z$ look 
similar to those of other growing transfer functions. 
These iterates provide the smooth transfer from the function 
$\mathrm{tra}$  to its inverse function $\mathrm{ArcTra}=\mathrm{tra}^{-1}$, and zeroth iterate 
is the identity function.

\begin{figure}[t]
\vskip 4pt
\sx{.7}{\begin{picture}(602,602)
\put(0,0){\ing{tranit}}
\put(311,590){\sx{2.5}{$y$}}
\put(311,495){\sx{2.4}{$2$}}
\put(311,395){\sx{2.4}{$1$}}
\put(311,295){\sx{2.4}{$0$}}
\put(307,194){\sx{2.4}{$-1$}}
\put(307,093.4){\sx{2.4}{$-2$}}

\put(083,308){\sx{2.4}{$-2$}}
\put(183,308){\sx{2.4}{$-1$}}
\put(297,308){\sx{2.4}{$0$}}
\put(397,308){\sx{2.4}{$1$}}
\put(497,308){\sx{2.4}{$2$}}
\put(590,308){\sx{2.5}{$x$}}

\put(96,531){\sx{2.4}{\rot{87}$n\!=\!10$\ero}}
\put(122,532){\sx{2.4}{\rot{87}$n\!=\!8$\ero}}
\put(156,532){\sx{2.4}{\rot{85}$n\!=\!6$\ero}}
\put(177,532){\sx{2.4}{\rot{84}$n\!=\!5$\ero}}
\put(204,532){\sx{2.4}{\rot{83}$n\!=\!4$\ero}}
\put(238,532){\sx{2.4}{\rot{82}$n\!=\!3$\ero}}
\put(287,532){\sx{2.4}{\rot{79}$n\!=\!2$\ero}}
\put(366,532){\sx{2.4}{\rot{71}$n\!=\!1$\ero}}
\put(448,528){\sx{2.4}{\rot{60}$n\!=\!0.4$\ero}}
\put(469,530){\sx{2.4}{\rot{57}$n\!=\!0.3$\ero}}
\put(491,532){\sx{2.4}{\rot{53}$n\!=\!0.2$\ero}}
\put(517,534){\sx{2.4}{\rot{49}$n\!=\!0.1$\ero}}

\put(541,531){\sx{2.4}{\rot{45}$n\!=\!0$\ero}}
\put(521,462){\sx{2.4}{\rot{37}$n\!=\!-0.2$\ero}}
\put(517,423){\sx{2.4}{\rot{30}$n\!=\!-0.4$\ero}}
\put(527,349){\sx{2.4}{\rot{17}$n\!=\!-1$\ero}}
\put(524,272){\sx{2.4}{\rot{8}$n\!=\!-2$\ero}}
\put(523,222){\sx{2.4}{\rot{6}$n\!=\!-3$\ero}}
\put(522,188){\sx{2.4}{\rot{4}$n\!=\!-4$\ero}}
\put(521,161){\sx{2.4}{\rot{3}$n\!=\!-5$\ero}}
\put(520,141){\sx{2.4}{\rot{2}$n\!=\!-6$\ero}}
\put(518,107){\sx{2.4}{\rot{1}$n\!=\!-8$\ero}}
\put(510,81){\sx{2.4}{\rot{1}$n\!=\!-10$\ero}}
\end{picture}}
\vskip -4pt
\hskip 14pt
\url{https://mizugadro.mydns.jp/t/index.php/File:TraitT.jpg}
\vskip -10pt
\caption{\large
$y\!=\!\mathrm{tra}^n(x)$ versus $x$ for various  $n$ by \rf{traite} \label{traiteplot}}
\vskip 9pt
\end{figure}

\section{Relation to other functions}
\baselineskip 18.9pt

Function SuTra can be expressed through function SuZex 
\be
\mathrm{SuTra}(z)=\mathrm{SuZex}(\ln(z))
\eL{SuTraSuZex}
and function AuTra can be expressed through function AuZex 
\be
\mathrm{AuTra}(z)=\exp(\mathrm{AuZex})
\eL{AuTraAuZex}
One can look at expansions of SuZex and AuZex from chapter 12 and see, that 
expansions for functions SuTra and AuTra can be obtained with relations \rf{SuTraSuZex} and \rf{AuTraAuZex}.
However, these representations  
may loss some precision, especially in the regions where the logarithm has low derivative,
and the exponential has high derivative. So, for the numerical tests, I use the special representations from the previous sections.
\vskip 2mm

In order to make the deep test of relations, suggested in this Book, the robust representations are required for all the functions involved.
For these reasons, the special representations and implementations for functions ArcTra, SuTra and AuTra are described above.
The readers are invited to make the asymptotical analysis for the relations above. Also, the numerical verification can be used.
For moderate values of the argument, the representation of SuTra through SuZex loss only few decimal digits in the precision.
I load the descriptions of functions mentioned in this section,\\
\url{https://mizugadro.mydns.jp/t/index.php/ArcZex} \\
\url{https://mizugadro.mydns.jp/t/index.php/AuZex} \\
\url{https://mizugadro.mydns.jp/t/index.php/SuZex}\\
\url{https://mizugadro.mydns.jp/t/index.php/ArcTra}\\
\url{https://mizugadro.mydns.jp/t/index.php/SuTra}\\
\url{https://mizugadro.mydns.jp/t/index.php/AuTra},\\
together with their complex double numerical implementations.
\vskip 2mm

On this I finish the consideration of the Trappmann function and its iterates. 
And at this point I finish consideration of examples of transfer functions, superfunctions, abelfunctions and the non-integer iterates.
Perhaps, I should explain, why I had spent so many efforts on this. I think, the best explanation is to remind the old folkloric story below.
\vskip 3mm

\baselineskip 15pt
Оne-legged friend of one Taylor asked him to sew the special one leg pants.
He payed well for the custom pants, but he needed also the pants for his dog, who, as himself, had lost one leg long time ago.Taylor sewed the pants for that 3-leg dog. The pants were beautiful, and friend of friend asked him the same for his normal, 4-leg dog..
The story is  long, the starfishеs and octopuses are mentioned there. En fin, the Taylor had elaborated tools to sew pantaloons for creatures with arbitrary number $n$ of legs.
And if tomorrow some extraterrestrials with $n$ legs come, the Taylor already has pantaloons for them.
\vskip 3mm
\baselineskip 18pt

I typed the story above in order to explain better, what is scientific research and what do the researchers. You may consider this as a kind of joke, 
but if someone needs to approximate, for example, the logarithmic function with entire functions, - then, like the Taylor mentioned, I already have such an approximation; it can be expressed through SuTra by formula  \rf{phin}. This may be considered as a small specific addition to various motivations
suggested in the Preface. 
\vskip 2mm

On this point I stop the main body of the Book and go to overview the results presented above.
This makes the content of the next chapter.


\chapter{~ Conclusion}
\baselineskip 19.6pt

\begin{wrapfigure}{r}{140px}
\begin{center}
\begin{picture}(124,40)
\put(0,0){\sx{.5}{\ing{oglavleni}}
}
\end{picture}
\end{center}
\vskip -17.8pt
\caption{\large %Читатель
\label{content}
\label{oglavleni}}
\end{wrapfigure}


This chapter discusses the same, as the Introduction.
The main difference of this chapter from the introduction is, that I assume, that the reader already has browsed at least some of the previous chapters,
as it is shown in figure \ref{oglavleni}, and understands sense of some notations used.
In order to show the need of conclusion, I remind the old legend.

\vskip 5mm

\baselineskip 15pt
\RU{
Один 
император 
решил изучить историю. Он поручил министерству науки разработать полный курс истории. Министры работали над таким учебником 
много лет. Наконец, во дворец 
приехал грузовик с заказанным тысячетомником. Император понял, что он за всю жизнь не сможет прочесть такой курс, и попросил его сократить. Ведущие историки работали над вторым изданием ещё несколько лет, и вот, в большом чемодане, императору был доставлен десятитомник. Но император к тому времени стал слаб глазами, и не мог осилить даже десятитомник. Историкам пришлось опять перерабатывать учебник, и через год к императору пришёл главный Историк и торжественно вручил ему брошюру "Краткий курс истории императорского двора". Император к тому времени уже заболел, и не мог прочесть даже эту брошюру. Он спросил Историка, можно ли этот курс переработать и сократить. Историк ответил: ``{\em Нового издания не надо. Я расскажу Вам прямо сейчас: Люди рождались, страдали и умирали.}''}
\EN{One
emperor
wanted to study history. He ordered the Ministry of Science to develop a full course of the world-wide history.
The greatest scientists were working on this tutorial during
many years. Finally, the heavy truck arrived with thousands volumes of "Complete Course of the World History".
The Emperor realised that all his life will not be sufficient to read this course.
The Emperor asker the President of the Ministry of Science to shorten the course.
The historians worked on the second edition during few years, and then, in a big pack,
"Trilogy of the World History" had been delivered to the Emperor.
But the Emperor already had weak eyes, and he could not read that Trilogy.
Again, the historians had to shorten the course.
A year later, the Top Historist came to the Emperor and gave him the pamphlet "A Brief History of the Imperial Family."
Emperor was old and ill, and could not even read that brochure. He asked Historist, whether the brochure can be reduced.
Historist answered: ``{\em No new edition is necessary. I'll tell you right now: People were born, suffered and died.}''
}%EN
\vskip 5mm

\large
\baselineskip 19pt
Several colleagues had told me, that this Book is too thick, and asked, if it can be shorter. 
As the historian in the story above, I follow the requests and describe the topic of superfunctions briefly.
Below, the main results of the Book are collected in a single section.
\section{Basic results}

\RU{Голоморфные функции можно итерировать. Какую попало голоморфную функцию $T$ можно обозвать ``передаточной функцией''.}%RU
\RU{Для передаточной функции $T$ можно построить суперфункцию как решение $F$ передаточного уравнения}%RU
\EN{Holomorphic functions can be iterated. The iterates can be expressed through the superfunction and the abelfunction;
then, the number of iterate has no need to be integer.}%EN
\vskip 2mm

In order to iterate some holomorphic function $T$, first, I declare it as ``transfer function''.
Then, I construct for the superfunction $F$, that is solution of the transfer equation
\be
F(z\!+\!1)=T\Big(F(z)\Big)  \nonumber
\ee
I need also the inverse function, $G\!=\!F^{-1}$. I call it ``Abel function'' or ``abelfunction''. It satisfies the Abel equation
\be
G\Big(T(z)\Big)=G(z)+1  \nonumber
\ee
When superfunction $F$ and the Abel function $G$ are established, the iterates of function $T$ can be expressed as
\be
T^n(z)=F\Big(n+G(z)\Big)  \nonumber
\ee
where the number $n$ of iterate has no need to be integer;
the transfer function can be iterated some non-integer, rational or even complex number of times.
However, for the integer $n$, the conventional expression of iterates holds:
\be
 {{T^n(z)}\atop{\phantom{sub}}} 
{{=}\atop{\phantom{sub}}}{{\underbrace{T\!\Big(T\!\big( .. T(z).. \big)\Big)}}
\atop{n ~\mathrm{evaluations~of~ function~} T}}
\nonumber
\ee
\vskip 2mm

Solution of the transfer equation is not unique. If some solution $F$ is found, then, one additional solution $f$ 
can be constructed with modification of the argument,
\be
f(z)=F\Big(z+\theta(z)\Big)
\nonumber\ee
where
$\theta$
is periodic holomorphic function with period unity. Accordingly, the new abelfunction
$g$ can be established.
Then, the new superfunction and the corresponding abelfunction will provide new, different iterates of the transfer function.
\vskip 2mm

Variety of superfunctions can be narrowed, if we establish, postulate the asymptotic behaviour of the superfunction in the complex plane. 
The superfunction with simple asymptotic behaviour I treated as principal.
Other superfunctions can be expressed with modification of argument of the principal superfunction.
Holomorphic periodic function grows at least exponentialy (as sin and cosine to in the direction of imaginary axis); even a small periodic modification is easy to reveal in the complex plane.
For this reason, for uniqueness, it is important to build-up the superfunctions for the complex argument,
even if they are supposed to be used for the real argument.
The criterion of holomorphism indicates, which of superfunctions is expected to have the physical sense and should be considered as ``true'' one.
\vskip 2mm

Some superfunctions and abelfunctions have special names.
These functions are collected in Table \ref{TableOfSuperfunctions}.
Some of them are widely known; one may use them without to know, that they are superfunctions.
\vskip 2mm

In principle, for any holomorphic function $F$, one can built-up the inverse function $G=F^{-1}$ and define
$T(z)=F\big(1+G(z)\big)$.
Then, such function $T$ can be treated as transfer function with known superfunction $F$, abelfunction $G$ and non-integer iterates
$T^n(z)=F\big(n+G(z)\big)$.
\vskip 2mm

The inverse problem, id est, construction of superfunction $F$ for some given transfer function $T$, is considered in this Book.
For this construction, the key question is about the fixed points of the transfer function, id est, about solutions $L$ of the 
equation
\be
L=T(L)
\nonumber\ee
As physicist, I am interested mainly in the real-holomorphic functions, for which $T(z^*)\!=\!T(z)^*$.
In order to reduce variety of solutions of the transfer equation, I postulate, that the superfunction approaches the fixed point $L$ 
at the infinity.
\vskip 2mm


Methods of asymptotic expansions of the superfunctions are suggested in the Book. At infinity, the superfunction is postulated to approach the fixed point of the transfer function. In many cases, this approach is exponential; and, in many cases, the exact superfunction appears as the limit at the multiple application of the transfer function to the asymptotic solution with displaced argument.
\vskip 2mm

It may happen, that all the fixed points $L$ of the transfer function $T$ are complex, not real. In particular, this is case of the natural exponent.
Then, the superfunction can be expressed through the Cauchy integral and solution of the corresponding integral equation.
Historically, the first complex map of superfunction of exponent, had been constructed with this representation.
Complex map of tetration is shown in figure \ref{tetmap} and at the First page of the cover of this Book.
Tetration tet appears as solution of the transfer equation
\be
\mathrm{tet}_b(z\!+\!1)= \mathrm e^{\mathrm{tet}_b(z)} ~,~
\mathrm{tet}_b(0)=1
\nonumber\ee
$\mathrm{tet}_b(z)$ is supposed to be bounded in the strip $|\Re(z)| \!\le\! 1$.
For real $b\!>\!1$ and real $x$, dependence $y\!=\! \mathrm{tet}_b(x)$ is shown in figure \ref{tet10bx}. 
Tetration can be constructed also for the complex base; the example with
$b\!=\!1.52598338517\!+\!0.0178411853321\, \mathrm i$ 
is shown in figure
\ref{sheldonmap}.
\vskip 2mm

It may happen, that the transfer function $T$ has no fixed point at all. One example of such a function is the Trappmann function
\be
T(z)=\mathrm{tra}(z)=z+\exp(z)
\nonumber\ee
However, even for this function, the superfunction can be constructed. It is called SuTra; its map is shoe in figure  \ref{sutramap}.
This is entire function with logarithmic asymptotic; up to my knowledge, before publication  \cite{sutra}, no one such function had been suggested.
\vskip 2mm



Since 2010, I claim, that I can construct the superfunction $F$, abelfunction $G$ and non-integer iterates for any
growing real-holomorphic transfer function $T$. This Book describes the sequence of attempts to negate, refute this claim.
All these attempts failed: I could not find any transfer function, for which I cannot construct the superfunction.
In support of my claim, the Book presents examples of transfer function with real fixed point(s), examples with complex fixed points, and the example of the transfer function without any fixed point.
While all the tests are successful, the ability of construction of superfunctions can be interpreted as a scientific fact.
\vskip 2mm

I just have mentioned, what is done. But, as usually, a lot of can be done about superfunctions.
The next section is dedicated to this.

\section{Future work}
\baselineskip 19.6pt
I tried to collect that I know about superfunctions and iterates, in this Book. However, always some fantasies remain, what else would be interesting to do. Some of these fantasies are collected below.
\vskip 2mm

The more efficient, more simple and more rigorous proofs of the existence and uniqueness of superfunction may require the future work, and also the additional conditions, that should be applied in order to provide the uniqueness of superfunction.
\vskip 2mm

Application of superfunctions into the laser science may be subject for the future analysis.
Especially, this refers to the laser science, where the physical sense of superfunctions and the non-integer iterates is especially explicit.
\vskip 2mm

The future work may be related to the specific case of superfunctions, namely, ackermanns.
Figure \ref{ackerplo} shows the graphics for the first five ackermanns:
addition of constant,
multiplication to constant,
exponent,
tetration and
pentation.
I think, similar graphics can be constructed also for higher ackermanns.
\vskip 2mm


The important suggestion for the future work is related with the automatisation of construction of superfunctions.
I mean the automatic algorithm, that begins with the transfer function, searches for the appropriate fixed points of this function
(if the transfer function has fixed points), chooses the appropriate asymptotic for the superfunction, use it to build-up the superfunction,
build-up the corresponding abelfunction and calculates the non-initeger iterates.
The software Mathematica already has name for such a procedure; it is called Nest.
Up to date of preparation of this Book, the routine Nest can deal only with very special case, namely, when the number of iterate is expressed with positive integer constant; in other cases, the 
call is interpreted as error.
The upgrade of that routine for the real and complex number of iterate would be interesting.

\vskip 2mm

Fantasies and curiosity should be motivations for any serious scientific research.
I collect the tools that can be used in this work. Many of them are described in this Book.

\section{Notations \label{Notations}}
\baselineskip 18pt

I try to use the same notations through the whole Book. In order to approach this, some notations are different from those used in the original publications. The notations are collected in tables
\ref{englishABC}, \ref{greekABC}.
I collect the most important notations, and those, that often cause confusions.

~

\newcommand \qq {$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$&$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$}
\RU{
\begin{table}[h]
\caption{Notations:Latin alphabet \label{russianABC}}
\vskip 4pt

\begin{tabular}{lr}
Абель Нильс Генрик \qq Изобретатель функции Абеля и уравнения Абеля \\
абель\qq приставка к имени функции для обозначения абельфункции\\
абельфункция \qq ~ ~ ~ ~ ~ ~ ~ решение уравнения Абеля, обратная фукция к суперфункции\\
Аскерманн \qq Изобретатель функций Аскерманна \\
aскерманн \qq  функция Аскерманна \\
Генрик Траппманн  \qq (Henryk Trappmann) мой критик, оппонент и соавтор  \\
Итерация номер $n$ \qq решение $f^n$ уравнений $f^{n+1}(z)=f(f^n(z))$,~ $f^1\!=\!f$ \\ 
Кнезер Хельмут \qq Изобретатель половинной итерации экспоненты  \\
Комплексная карта \qq представление голоморфной функции изолиниями\\
\qq её вещественной и мнимой частей (или изолиниями её модуля и фазы)\\
Корень из факториала  \qq  $\mathrm{Factorial}^{1/2}(z)$, решение $h$ уравнения $h(h(z))=z!$ \\
Обращение (функции) \qq нахождение обратной функции, InverseFunction\\
Передаточная функция  \qq  зависимость выхода усилителя от входa,\\
                                          \qq функция $T$ в передаточном уравнении\\
Передаточное уравнение \qq  $F(z\!+\!1)=T(F(z))$ \\
Супер \qq приставка для обозначения её суперфункции \\
Суперфункция \qq решение $F$ передаточного уравнения $F(z\!+\!1)\!=\!T(F(z))$ \\
Тетрация \qq решение  tet уравнений $\mathrm{tet}(z\!+\!1)\!=\!\exp(\mathrm{tet}(z))$ ,~  $\mathrm \tet(0)\!=\!1$\\
Тетрация по основанию $b$ \qq решение  уравнений $\mathrm{tet}_b(z\!+\!1)\!=\!b^{\mathrm{tet}_b(z)}$,~  $\mathrm \tet(0)\!=\!1$\\
Траппманн Генрик   \qq (Henryk Trappmann) мой критик, оппонент и соавтор  \\
Уравнение Абеля \qq $G(T(z))=G(z)+1$ \\
Усилитель  \qq  фильтр, в котором выход может быть больше входа \\
Функция Абеля  \qq  абельфункция. Обратная функция от суперфункции \\
Функция Дойя \qq $\mathrm{Doya}(z)=\mathrm{LambertW}(z \mathrm e^{z+1})$, см. также формулу \rf{Doya} \\
Функция Келлера \qq $\mathrm{Keller}(z)=z+\ln\!\Big(\mathrm e- \mathrm e ^{-z} (\mathrm e\!-\!1)\Big)$, см. формулу \rf{KellerDef}  \\
Функция Таня \qq  Решение уравнения $\mathrm{Tania}'(z)=\frac{\mathrm{Tania}(z)}{1+\mathrm{Tania}(z)}$ \\
Фукция Шока \qq  $\mathrm{Shoka}(z)=z+\ln\!\Big(\mathrm e ^{-z}\!+\! \mathrm e\!-\!1\Big)$, см. формулу \rf{Shoka}  \\
Фильтр  \qq  физическая имплементация передаточной функции \\
\end{tabular}
\end{table}
}%RU
\begin{table}[h]
\caption{Notations, Latin alphabet \label{englishABC}}
\vskip 1mm
\begin{tabular}{lr}
$\mathcal A(m,z)=\mathcal A\Big(m\!-\!1,\mathcal A(m,z\!-\!1)\Big) \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!  \!\!\!\!\!\!\!\!\!\!$ \qq %канонический аскерманн,
canonical ackermann \rf{ackerbi}\\
$A_{b,n}$ \qq 
ackermann to base $b$ of order $n$, \rf{asa3}\\
$a_n$ \qq 
Coefficients in various asymptotic expansions\\
$\mathrm{AuExp}_b=\mathrm{SuExp}_b^{-1}$ & growing abelexponent to base $b$ \\
AuTra \qq Abel function for the Trappmann function\\
AuSin \qq 
abelsine\\
$\mathrm{Arc}$ \qq 
prefix indicating the inverse function\\
$b$ \qq base of ackermanns, \rf{tetb}\\
$c_n$ \qq coefficients in expansions, for ex., \rf{e1egpm} or \rf{arctrap1} \\
$C_n$ \qq %Коэффициенты в разложении 
coefficients in expansions, for ex., \rf{GC}\\
$\mathrm e=\exp(1)\approx 2.7$ \qq %основание натуральных логарифмов\\
base of natural logarithms\\
$\exp$ \qq 
exponent to base
$\mathrm e$ \\
$\exp_b$ \qq 
exponent to base 
$b$ \\
$F$ \qq 
Name, used to denote superfunctions\\
$\mathrm{Factorial}(z)=z!$ \qq 
holomorphic solution of equations 
$z!=z\, (z\!-\!1)!$\\
$f$ \qq %имя, используемое для каких попало функций\\
name used for various functions\\
$G=F^{-1}$ \qq name used to denote abelfundtions\\
$h\!=\!\mathrm{Elu}_s^{0.5}$
\qq  half iterate of logistic operator \\
$\mathrm i=\sqrt{-1}$ \qq 
imaginary unit
$\mathrm i^2=-1$ \\
$i$ \qq variable, usually it takes integer values\\
$K$ \qq 
plug for the contour integral rf{K}, or $\mathrm e^k$
\\
$k$ \qq %инкремент асимптотического представления суперфункции\\
increment in the asymptotic representation of superfunction \rf{Lea2}\\
$L$ \qq 
fixed point of transfer function $T$, id est, solution of  ~ $T(L)\!=\!L$ \\
$\ln$ \qq %Логарифм по основанию 
logarithm to base $\mathrm e$\\
$\log_b$ \qq 
Logarithm to base $\mathrm b$\\
$M$ &  %количество термов в оборванных сериях\\
number of terms in truncated series\\
$O$ \qq  %функция, которая растет не быстрее своего аргумента  \\
some function that grows-up not faster than its argument\\
$P$ \qq \hskip -28pt % период или асимптотический период; функция, обратная к $Q$\\
period or asymptotic period; also is used as inverse of function $Q$\\
$Q$ &  \hskip -24pt  %Используется как функция, обратная к $P$ \\ 
is used as inverse function of $P$ in table \ref{TableOfSuperfunctions}\\
$\mathrm{Super}$ \qq  %приставка, указывающая на её суперфункцию\\
prefix to name of function, indicating its sperfunction\\
$s$ & %Параметр логистического оператора 
parameter of the logistic operator \rf{logisticop} \\
$T$ & %Передаточная функция\\
transfer function\\
$\mathrm{tet}$ &  %натуральная тетрация 
natural tetration \rf{tet}, \cite{analuxp}\\
$\mathrm{tet}_b$ \qq  %решение $f$ уравнения $f(z\!+\!1)=b^{f(z)}$, см. рисунок \ref{tet10bx} \\
solution $f$ of equaitons $f(z\!+\!1)\!=\!b^{f(z)}$,~  $f(0)\!=\!1$, see fig. \ref{tet10bx} \\
$u$ \qq %вещественная часть функции на комплексных картах \\
real part of function in the complex map\\
$v$ \qq %мнимая часть функции на комплексных картах \\
imaginary part of function in the complex map\\
$x$ \qq %используется в качестве немого аргумента как вещественное число\\
used as dummy parameter, that takes real values\\
$y$ \qq %используется в качестве немого аргумента\\
used as dummy parameter, that takes real values\\
$z$ \qq %используется в качестве немого аргумента как комплексное число\\
used as dummy parameter, that takes complex values; often $z\!=\! x \!+\!\mathrm i y$ \\
\end{tabular}
\vskip -200pt
\end{table}
\newpage

\begin{table}
\caption{Notations, Greek alphabet \label{greekABC}}
\vskip 3mm
\begin{tabular}{lr}
$\alpha$ , $\beta$ , $\gamma$  & %используются как коэффициенты в разложениях \\
used as parameters in the expansions\\
$\eta\!=\!\exp(1/\mathrm e)\!\approx\! 1.444667861 \!\!\!$ \qq 
Henryk base\\
$\varepsilon=\exp(kz)$ & %малый параметр в разложении суперфункции\\
small parameter in the expansion of superfunctions\\
$\theta$ & %голоморфная периодичная функция с периодом единица  \\
holomorphic periodic function with period unity\\
$\zeta$ & $\ds \zeta(z)\!=\! \sum_{n=1}^\infty \frac{1}{n^z}$ ,  
Riemann function
\rf{u2u3zeta},  \cite{gr,IRENE}\\
$\Omega$ & 
contour of integration in the Cauchy formula
\rf{Cauchy}\\ ~\\~
\end{tabular}
\end{table}
\section{Afterwards} 
\baselineskip 18pt
The first (Russian) version of this Book happened thicker, than I had expected. 
It could be much thicker; because all the time, there is the evil illusion, that some additional {\small small formula}
should greatly simplify the understanding. However, with each formula from the book, several additional pictures can be plotted.
However, I think, for the Reader, it will be much better, to plot some picture, than to see the gallery of similar figures in the Book.
\vskip 6pt

I tried to make this Book shorter, than its first Russian version. Actually, the Book happened to be longer, thicker, because I include here the chapter about the Nemtsov function
$\mathrm{Nem}_q(x)\!=\!x\!+\!x^3\!+\!qx^4$; this is important example of the exotic iteration of transfer function with the specific expansion at the fixed point.
\vskip 2mm

Some things are still dropped out from the Book.
I did not include the holomorphic extension of tetration beyond the cut lines.
I did not include figures to iterates of the exponent to complex base.
And I did not include many other figures, assuming, that the Readers can download the generators of the figures and plot all modifications they need.
\vskip 6pt

The Readers are invited to download algorithms, figures and their generators from 
\url{https://mizugadro.mydns.jp/t/index.php/Category:Book}; with these tools, the colleague may continue from that points of research, where I am now.
I believe, this is correct style of making science, where all the results are available for researchers. This is supposed to simplify the verification and refutation
(if anything is seriously wrong), as well as revealing and correction of mistakes in formulas and bugs in the algorithms.
I invite the colleagues to expose their results in a similar way.


\newpage

{
\begin{figure}
\begin{center}
\sx{.34}{\begin{picture}(640,470)
\put(0,0){\ing{DevchonkaRoux}}
\end{picture}}
\end{center}
\vskip -18pt
\caption{Near relatives, whose existence is essential for writing of this Book \label{DevchonkaRoux}}
\end{figure}}%JP

\section{Acknowledgement}
\baselineskip 17.9pt

I am grateful to relatives (Figure \ref{DevchonkaRoux}) for the tolerance with respect to this Book, it took much more time and efforts, than expected.
\vskip 2pt

\begin{wrapfigure}{r}{180px}
\sx{.266}{\begin{picture}(800,490)
\put(0,-40){\ing{bookwormbw}}
\end{picture}}
\caption{
Bookorm \cite{bookworm}
\label{bookworm}}
\end{wrapfigure}

I am grateful to colleagues, who helped me to collect the literature on superfunctions (Figure \ref{bookworm}) 
and arrange the server, database and mediawiki: without these instruments, I would get lost among a thousand files used for generation of this Book. 

The goal was to get possibility to answer the questions on superfunctions with either {\em Nobody knows this!} or
{\em Das ist in meinem Buch!} (see figure \ref{ainu}). Perhaps, this goal is not reachable; the new algorithms 
are expected to be reported \cite{paulsen}. Tanks to colleagues who keep doing superfunctions.

\begin{figure}[h]
\begin{center}
\vskip 8pt
\sx{.12}{\begin{picture}(2100,1500)
\put(0,0){\ing{ainu}}
\end{picture}}
\end{center}
\vskip -24pt
\caption{Das ist in meinem Buch! \iL{ainu}}
\vskip -24pt
\end{figure}

\chapter{~ Supplement \label{supplement}}
\baselineskip 21pt

\section{About the cover}
\begin{picture}(0,230)
\put(0,0){\sx{.28}{\ing{tetma050}}}
\end{picture}
\vskip `8pt

At the front cover of this book, the map is shown,
from \url{https://mizugadro.mydns.jp/t/index.php/File:Tetma.jpg}
\vskip 2mm

This is one of versions of figure \ref{tetmap}; 
Some details and labels are removed in order to simplify the aesthetic view.
\vskip 4mm

The cover of the Russian version is loaded as \url{https://mizugadro.mydns.jp/t/index.php/File:Covervi.jpg}
\vskip 3mm


\section{Maple and tea \label{MapleAndTea}}
\baselineskip 16.7pt
Here is the company C of computers; they master:\\
Per each two years, their PCs run twice faster.
\vskip 1mm

Here is company S of the soft; they work hard as plowers,\\
So, each new release runs 0.7 times slower;\\
The soft is to run at the user's PC,\\
Which is made, of course, by the company C.
\vskip 1mm

Here is user U, buys from C and from S,\\
the newest model and the latest release.\\
U presses some key, say, key number K;\\
The soft S responds, with some delay,\\
during some time, say, during time t,\\
to let user U to have some tea.
\vskip 1mm

How many releases per year does sell S to U\\
to let him have tea, while he has nothing to do,\\
just waiting response by the soft to key K,\\
keeping the same time t of delay,\\
neutralizing the efforts of company C\\
to run faster U's task at his modern PC?
\vskip 1mm

This problem above is not correct, because\\
The tea-concern, together with C, of course,\\\
support the efforts of the company S\\
to make even bigger the newest release,\\
To force U buy more tea and a newest PC,\\
to boost business of C and the concern of tea.
\vskip 2mm

(2007)

\begin{figure}[t]
\sx{.52}{\begin{picture}(360,390)
\put(-12,0){\ing{MaplePlotLog02}}
\end{picture}}
\sx{.52}{\begin{picture}(400,390)
\put(0,0){\ing{MapleImpPlotErf02}}
\end{picture}}
\vskip -16pt
\caption{The reader should have Maple-10 installed, in order to reproduce the plots above with the codes supplied at the top of each picture \label{figmaple}}
\end{figure}

\newpage
\begin{figure}[h]
\vskip -16pt
\sx{1.3}{\begin{picture}(328,412)
\put(10,9){\includegraphics{ausintay40}}
\put(2,400){\sx{1.2}{$y$}}
\put(2,306){\sx{1.2}{$1$}}
\put(2,206){\sx{1.2}{$0$}}
\put(-7,106){\sx{1.2}{$-1$}}
\put(9,-1){\sx{1.2}{$0$}}
\put(109,-1){\sx{1.2}{$1$}}
\put(157,-1){\sx{1.2}{$\pi/2$}}
\put(209,-1){\sx{1.2}{$2$}}
\put(309,-1){\sx{1.2}{$3$}}
\put(319,-1){\sx{1.2}{$x$}}
\put(184,208){$v\!=\!0$}
\put(98,200){\rot{90}$u\!=\!2$\ero}
\put(114,200){\rot{90}$u\!=\!1$\ero}
\put(147,186){\rot{41}$u\!=\!0$\ero}
\put(136,131){\rot{-40}$u\!=\!-1$\ero}
\put(125,93){\rot{23}$v\!=\!0$\ero}
\put(92,288){\rot{66}$u\!=\!-2$\ero}
\put(131,282){\rot{31}$u\!=\!-1$\ero}
\put(148,260){$u\!=\!-0.5$}
\put(86,158){\rot{90}$v\!=\!3$\ero}
\put(100,155){\rot{90}$v\!=\!2$\ero}
\put(122,148){\rot{90}$v\!=\!1$\ero}
\put(159,144){\rot{90}$v\!=\!0.2$\ero}
\put(172,144){\rot{90}$v\!=\!0$\ero}
\put(186,138){\rot{90}$v\!=\!-0.2$\ero}
\put(340,30){\rot{90}\sx{.9}{\url{https://mizugadro.mydns.jp/t/index.php/File:Ausintay40t50.jpg}}\ero}
\end{picture}}
\vskip -10pt
\caption{Map of the Taylor expansion $f$ of AuSin with 40 terms \label{ausintay40} in the standard notations.
$u\!+\!\mathrm i v=f(x\!+\!\mathrm i y)$. \label{ausinte}
} \label{ausintee}
\vskip -90pt
\end{figure}

\rule{0pt}{70pt}

\section{Taylor expansion of  AuSin}
\baselineskip 17pt

In chapter 13, the Taylor expansion of function AuSin at  $\pi/2$ is mentioned, \rf{ausintaylor}. 
In figure \ref{ausintay40}, the complex map of the truncated series with 40 terms is shown. 
This map should be compared to Figure
\ref{ausinmap}. 
Such an expansion can be used to boost the precise evaluation of AuSin for values of the argument in vicinity of $\pi/2$. Also,
coefficients of this expansion can be used for evaluation of coefficients of expansion of SuSun at zero; for example, wight routine InverseSeries.
\newpage

\section{Sites \label{SupplementSites}}
\baselineskip 16pt
Goal to his Book is not to attract attention of Reader to the cites below,
but to provide the description of superfunctions, that releases the reader from need to browse the sites below.
\vskip 2mm

{\normalsize
\baselineskip 14pt
\url{https://allmybase.com/dropbox/tetration.pdf}

https://cdn.bitbucket.org/bo198214/bunch/downloads/main.pdf H.Trappmann, D.Kouznetsov. 5+ methods for real analytic tetration. June 28, 2010. (Since 2020, does not open)

\url{https://en.citizendium.org/wiki/Cauchy-Riemann_equations}

\url{https://en.citizendium.org/wiki/Superfunction}

\url{https://en.citizendium.org/wiki/Tetrational}

\url{https://en.citizendium.org/wiki/Holomorphic_function}

\url{https://en.wikipedia.org/wiki/Abel_equation}

\url{https://en.wikipedia.org/wiki/Abel_function}

\url{https://en.wikipedia.org/wiki/Cauchy's_integral_formula}

\url{https://en.wikipedia.org/wiki/Hellmuth_Kneser}

\url{https://en.wikipedia.org/wiki/Niels_Henrik_Abel}

\url{https://en.wikipedia.org/wiki/Superfunction}

\url{https://en.wikipedia.org/wiki/Tetration}

\url{https://math.eretrandre.org/tetrationforum/index.php}

\url{https://math.stackexchange.com/tags/tetration}

\url{https://mathworld.wolfram.com/Tetration.html}


\url{https://oeis.org/wiki/Tetration}

\url{https://mizugadro.mydns.jp/t/index.php/Abel_function}

\url{https://mizugadro.mydns.jp/t/index.php/ArcShoka}

\url{https://mizugadro.mydns.jp/t/index.php/AuSin}

\url{https://mizugadro.mydns.jp/t/index.php/AuTra}

\url{https://mizugadro.mydns.jp/t/index.php/AuZex}

\url{https://mizugadro.mydns.jp/t/index.php/Complex_map}

\url{https://mizugadro.mydns.jp/t/index.php/Doya_function}

\url{https://mizugadro.mydns.jp/t/index.php/Factorial}

\url{https://mizugadro.mydns.jp/t/index.php/Holopmorphic_extension_of_Collatz_Subsequence}

\url{https://mizugadro.mydns.jp/t/index.php/Keller_function}

\url{https://mizugadro.mydns.jp/t/index.php/LambertW}

\url{https://mizugadro.mydns.jp/t/index.php/Logistic_sequence}

\url{https://mizugadro.mydns.jp/t/index.php/Regular_iteration}

\url{https://mizugadro.mydns.jp/t/index.php/Shoka_function}

\url{https://mizugadro.mydns.jp/t/index.php/Superfunction}

\url{https://mizugadro.mydns.jp/t/index.php/Superfactorial}

\url{https://mizugadro.mydns.jp/t/index.php/SuSin}

\url{https://mizugadro.mydns.jp/t/index.php/SuTra}

\url{https://mizugadro.mydns.jp/t/index.php/SuTra}

\url{https://mizugadro.mydns.jp/t/index.php/Table_of_superfunctions}

\url{https://mizugadro.mydns.jp/t/index.php/Tania_function}

\url{https://mizugadro.mydns.jp/t/index.php/Tetration}

\url{https://www.proofwiki.org/wiki/Definition:Superfunction}

\url{https://www.proofwiki.org/wiki/Definition:Tetration}

\url{https://www.tetration.org/Tetration/index.html}  D.Geisler. What lies beyond exponentiation? %2012 

\url{https://www.youtube.com/watch?v=z-mfxP1Tmfw}  Kasane Teto. Tetration $\uparrow\uparrow$. 2012. %по-японски
\vskip -18pt
}


\section{New notations \label{NewNotations}}
\baselineskip 18.2pt
In future, the new notations will be requested. Below, I suggest tentative names for them:
\vskip 1mm

{\bf %Трансферирование 
Transferation}.
For given function $f$, construction of the transfer function
$T(z)\!=\!f\big(1\!+\!f^{-1}(z)\big)$.
For this transfer function $T$, function $f$ is superfunction and $f^{-1}$ is abelfunction.
\vskip 1mm

{\bf %Трансфирование} (
Transation}. %Для данной функции $f$, построение предаточной функции 
For given function $f$, construction of the transfer function
$T(z)\!=\!f^{-1}\big(1\!+\!f(z)\big)$.
For this transfer function $T$, function $f^{-1}$ is superfunction and $f$ is abelfunction.
\vskip 1mm

{\bf %Суперирирование} (s
Superation}. %Для данной функции $f$, построение суперфункции $F$ как решения 
For given function $f$, construction of superfunction $F$ as solution of equation
$F(z\!+\!1)\!=\! f\big(F(z)\big)$. 
In this case, $f$ appears as transfer function.
\vskip 1mm

{\bf %Супирование} (s
Supation}. %Для данной функции $f$, построение абельфункции $G$ как решения  уравнения Абеля 
For given function $f$, construction of abelfunction $G$ as solution of the Abel equation
$G\big(f(z)\big)\!=\!G(z)\!+\!1$. In this case also $f$ appears as a transfer function.
\vskip 1mm

\begin{wrapfigure}{l}{182pt}
\normalsize
\vskip -4pt
\sx{1.06}{\begin{picture}(170,180)
\put(10,172){\sx{1.4}{$F$}}
\put(10,28){\rot{90}\sx{1.2}{$\leftarrow$ transferation $\leftarrow$}\ero}
\put(23,30){\rot{90}\sx{1.2}{$\longrightarrow$ superation $\longrightarrow$}\ero}
\put(6,4){\sx{1.4}{$T$}} \put(150,6) {\sx{1.4}{$G$}}
\put(32,13){\sx{1.2}{$\longrightarrow$ supation $\longrightarrow$}}
\put(28,0){\sx{1.2}{$\leftarrow$ transfation $\leftarrow$}}
\put(38,160){\rot{-49}\sx{1.19}{$\longleftarrow$ ~ InverseFunction ~ $\longrightarrow$}\ero}
\end{picture}}
\vskip -2pt
\caption{ \large $T$, $F$ and  $G\!=\!F^{-1}$ 
\label{figTFG}}
\vskip -38pt
\end{wrapfigure}

Transferation is inverse operation with respect to suppuration.
Transfation is inverse operation with respect to supation.
This is shown in figure  \ref{figTFG} for the transfer function $T$, superfunction $F$ and abelfunction $G$.
\vskip 2mm

This Book is dedicated to superation and supation,
and also to the additional requirements, that provide the uniqueness of superfunction $F$ and abelfunction $G$.
\vskip 2mm

In this Book, I do not use the new words shown in figure \ref{figTFG}; while, there is no need to use them.
In the similar way, there was no need in the special terms for differentiation and integration, until these operations became routines.
However, the terms shown in figure \ref{figTFG} will be requested as soon as the automatic construction of superfunctions and abelfunctions will be realised.
In language Mathematica, for the superation, there exist name Nest (until year 2017, this routine is supported only for natural values of the number of iterate).
\vskip 2mm

I am not sure, which names from figure \ref{figTFG} will be usual. 
For this reason, I do not use these names in the Book. However, the operations, mentioned in figure \ref{figTFG} exist; they will require some names.


\section{Extended abstract}
\Large 
\baselineskip 23pt

The Book is dedicated to construction of superfunction $F$ and abelfunction $G$ for given transfer function $T$.
The formalism of superfunctions is considered with examples, that are presented in ready-to-use form.
The reader is supposed to know something about the complex numbers and elementary functions.
 
\vskip 4mm

The superfunction is solution of  the transfer equation 

$F(z\!+\!1)=T\big(F(z)\big)$

The abelfunction $G=F^{-1}$ satisfies the Abel equation 

$G\big(T(z)\big)=G(z)+1$

In order to provide the uniqueness of solution, the additional requirements on $F$ are applied,
referring to its behaviour in the complex plane.
\vskip 4mm

The $n$th iterate of function 
$T$
is denoted with superscript:
\be
 {{T^n(z)}\atop{\phantom{sub}}} {{=}\atop{\phantom{sub}}}{{\underbrace{T\!\Big(T\!\big( .. T(z).. \big)\Big)}}\atop{n ~\mathrm{evaluations~of~ function~} T}}
\nonumber
\ee
With superfunction $F$ and  abelfunction $G$, the iterate is expressed as ~
$T^n(z)=F\big(n\!+\!G(z)\big)$ . 
In this representation, the number $n$ of iterate has no need to be integer.
\vskip 4mm

Examples of superfunctions are considered and collected as Table \ref{TableOfSuperfunctions}.
Superfunctions are constructed for sin, factorial, exponential, tetration and other functions.

Many explicit plots and complex maps for these functions, superfunctions, and iterates are included. 
The figures are loaded also to TORI together with their generators at

\sx{.98}{\url{https://mizugadro.mydns.jp/t/index.php/Category:Book}}
\vskip 4mm

The formalism of superfunctions greatly extends the set of functions available for applications in the scientific research.


\section{Keywords \label{keywords}}

{\large In this section, I suggest the essence of the notations}
\Large
\baselineskip 36pt

$T$ ~ ~ ~ ~ ~ Transfer function

$T\big(F(z)\big)=F(z\!+\!1)$ ~ ~ ~ Transfer equation, superfunction

$G\big(T(z)\big)=G(z)+1$ ~ ~ ~ Abel equation, abelfunction

$F\big(G(z)\big)=z$ ~ ~ ~ ~ ~ Identity function

$T^n(z)=F\big(n+G(z)\big)$ ~ ~ ~ $n$th iterate
\vskip 8pt

$\ds F(z)=\frac{1}{2\pi \mathrm i} \oint \frac{F(t) \, \mathrm d t}{t-z}$ ~ ~ ~ Cauchy integral
\vskip 8pt

$\mathrm{tet}_b(z\!+\!1)=b^{\mathrm{tet}_b(z)}$ ~ ~ ~ tetration to base $b$

$\mathrm{tet}_b(0)=1$ ~, ~ ~  $\mathrm{tet}_b\big(\mathrm{ate}_b(z)\big)=z$

$\mathrm{ate}_b(b^z)=\mathrm{ate}_b(z)+1$ ~ ~ arctetration to base $b$

$\exp_b^{~n}(z)=\mathrm{tet}_b\big(n+\mathrm{ate}_b(z)\big)$ ~ ~ $n$th iterate of function ~ $z\!\mapsto\! b^z$
\vskip 8pt

$\ds \mathrm{Tania}^{\prime}(z)=\frac{\mathrm{Tania}(z)}{\mathrm{Tania}(z)\!+\!1}$ ~ ~   Tania function,~  $\ds \mathrm{Tania}(0)\!=\!1$ 
\vskip 8pt

$\ds \mathrm{Doya}(z)=\mathrm{Tania}\big(1\!+\!\mathrm{ArcTania}(z)\big)$ ~ ~   Doya function

$\ds \mathrm{Shoka}(z)=z+\ln(\mathrm e^{-z}\!+\!\mathrm e \!-\! 1)$ ~ Shoka function

$\ds \mathrm{Keller}(z)=\mathrm{Shoka}\big(1\!+\!\mathrm{ArcShoka}(z)\big)$ ~ ~   Keller function

$\ds \mathrm{tra}(z)=z+\exp(z)$ ~ ~ ~  Trappmann function

$\ds \mathrm{zex}(z)=z\,\exp(z)$ ~ ~ ~ ~ Zex function

\large
\baselineskip 20pt

\begin{thebibliography}{99} \label{references}
\baselineskip 17.8pt
\bibitem{Rabelais} %\begin{picture}(0,0) \put(-30,50){\rule{300pt}{30pt}} \end{picture}
\url{https://en.wikisource.org/wiki/Gargantua/The_Author%27s_Prologue_to_the_First_Book}
Gargantua by François Rabelais, translated by Thomas Urquhart and Peter Antony Motteux.
{\em 
..Most noble and illustrious drinkers, and you thrice precious pockified blades (for to you, and none else, do I dedicate my writings),
}

\bibitem{moliere}
\url{https://moliere-in-english.com/bourgeois.html} J.B. Moliere. The Bourgeois Gentleman. (1670)
English adaptation by Timothy Mooney. Act Two, Scene Four. {\em .. These forty years now, I've been speaking in prose without knowing it! ..} 

\bibitem{abel}
\url{https://eudml.org/doc/183045}
N.H.Abel. "Untersuchung der Functionen zweier unabhängig ver\"anderlichen Gr\"ossen $x$ und $y$, wie $f(x, y)$, welche die Eigenschaft haben, dafs $f(z,f(x,y))$ eine symmetrische Function vom $z,x$ und $y$ ist.. Journal f\"ur die reine und angewandte Mathematik, V.1 (1826)  Z.11–15. %cite03

\bibitem{xutore} 
{\normalsize \url{https://xixvek.wordpress.com/2013/03/27/translation-comparison-evenings-at-a-farmhouse-near-dikanka}}
N.V.Gogol. Evenings at a Farmhouse near Dikanka. MARCH 27, 2013. 
{\em “What oddity is this: Evenings on a Farm near Dikanka? What sort of Evenings have we here? And thrust into the world by a beekeeper! God protect us! As though geese enough had not been plucked for pens and rags turned into paper! As though folks enough of all classes had not covered their fingers with inkstains! The whim must take a beekeeper to follow their example! Really there is such a lot of paper nowadays that it takes time to think what to wrap in it.”}

\bibitem{gogol}
\url{https://archive.org/stream/deadsoulsgog00gogouoft/deadsoulsgog00gogouoft_djvu.txt}
THE WORKS OF NIKOLAY GOGOL. DEAD SOULS. VOL. I. Translated by Constance Qarnett.. Digitized by the Internet Archive in 2007
{\em .. Happy is the traveller who, after a long and weari-some journey.. }
\JP{%\small 
\renewcommand\baselinestretch{.2}
\baselineskip 1pt %\em 
.. Happy is the traveller who, after a long and weari-some journey with its cold and sleet, mud, postingstation superintendents waked out of their sleep, jingling bells, repairs, disputes, drivers, blacksmiths and all sorts of rascals of the road, sees at last the familiar roof with its lights flying to meet him. And there rise before his mind the familiar rooms, the delighted outcry of the servants running out to meet him, the noise and racing footsteps of his children, and the soothing gentle words interspersed with passionate kisses that are able to efface everything gloomy from the memory. Happy the man with a family and nook of his own, but woe to the bachelor ! \\
Happy the writer who, passing by tedious and repulsive characters that impress us by their painful reality, attaches himself to characters that display the loftiest virtues of humanity, who, from the great whirlpool of human figures flitting by him daily, has selected only the few exceptions, who has never tuned his lyre to a less exalted key, has never stooped from his 
pinnacle to his poor, insignificant fellow-creatures, but without touching the earth has devoted himself entirely to his elevated images that are utterly remote from it. His fair portion is doubly worthy of envy ; he lives in the midst of them as in the midst of his own family ; and, at the same time, his fame resounds far and wide. He clouds men's vision with enchanting incense; he flatters them marvellously, covering up the gloomy side of life, and exhibiting to them the noble man. All run after him, clapping their hands and eagerly following his triumphal chariot. \\
They call him a great world-famed poet, soaring high above every other genius as the eagle soars above the other birds of the air. Young ardent hearts are thrilled at his very name ; responsive tears gleam in every eye. . . . No one is his equal in power — he is a god ! But quite other is the portion, and very different is the destiny of the writer who has the temerity to bring to the surface what is ever before men's sight and  is unseen by their indifferent eyes — all the terrible overwhelming mire of trivialities in which our life is entangled, all that is hidden in the often cold, petty everyday characters with which our bitter and dreary path through life swarms, and with the strong hand of a relentless sculptor dares to present them bold and distinct to the gaze of all ! It is not for him to receive the applause of the people, it is not his lot to behold the grateful tears and single-hearted rapture of souls stirred by his words ; no girl of sixteen, with her head turned, flies to meet him with heroic enthusiasm. It is not his lot to lose himself in the sweet enchantment of sounds he has himself evoked. And lastly, it is not his lot to escape the contemporary critic, the hypocritical callous contemporary critic, who will call his cherished creations mean and insignificant, will assign him an ignoble place in the ranks of writers who have insulted humanity, will attribute to him the qualities of his heroes, will strip him of heart and soul and the divine fire of talent. For the contemporary critic does not recognise that the telescope through which we behold the sun and the microscope which unfolds to us the movements of unnoticed insects are equally marvellous. For the contemporary critic does not recognise that great spiritual depth is needed to light up a picture of ignoble life and transform it into a gem of creative art. For the contem-porary critic does not admit that the laughter of lofty delight is worthy to stand beside exalted lyrical emotion, and that there is all the world of difference between it and the antics of clowns at a fair ! All this the critic of to-day does not admit, and he will turn it all into the censure and dishonour of the unrecognised writer. Without sympathy, without response, without compassion, he is left by the roadside like the traveller without a family. Hard is his lot and bitterly he feels his loneliness. ..}%JP

\bibitem{fatou}
\url{https://www.numdam.org/item?id=BSMF_1919__47__161_0}\\
 P. Fatou. Sur les équations fonctionnelles. Bulletin de la Société Mathématique de France, 47 (1919), p. 161-271. %cite06
 
 \bibitem{acker}
\url{https://www.springerlink.com/content/px2g58k5j3q45535/}\\
Wilhelm Ackermann. Zum Hilbertschen Aufbau der reellen Zahlen.  %MATHEMATISCHE ANNALEN
Mathematische Annalen {\bf 99}, Number 1(1928), Z.118-133. %cite07

 \bibitem{bulga} 
\url{https://www.masterandmargarita.eu/estore/pdf/eben001_mastermargarita_glenny.pdf}
Mikhail Bulgakov. The Master and Margarita. Translated from the russian by Michael Glenny. Collins and Harvill Press, London, 1967.
Chapter 24. The Master is Released.  .. {\em Manuscripts don't burn.} ..

\bibitem{boy}
\url{https://en.wikipedia.org/wiki/File:Boy_on_snow_sled,_1945.jpg} Father of JGKlein. Boy on snow sled, 1945. %cite09

\bibitem{kneser}
\url{https://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851} ~
\url{https://www.ils.uec.ac.jp/~dima/Relle.pdf} 
Hellmuth Kneser. Reelle analytische L\"osungen der Gleichung $\varphi(\varphi(x))=e^x$. Equationes Mathematicae, Journal fur die reine und angewandte Mathematik {\bf 187} 56–67 (1950) %cite10

\bibitem{szekeres}
\url{https://link.springer.com/article/10.1007\%2FBF02559539} 
G.Szekeres. Regular iteration of real and complex functions. Acta Mathematica
1958, Volume 100, Issue 3-4, pp 203-258. %cite11

\bibitem{wright} \url{https://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183523043}
E.M.Wright. Solution of the equation $ze^z=a$. Bulletin of the American Mathematical Society {\bf 65}, p.89-93 (1959). %cite12

\bibitem{gr}
\url{https://www.mathtable.com/gr/} Gradshteyn and Ryzhik. Table of Integrals, Series, and Products. (Many editions)

\bibitem{ponedelnik} %cite14
\url{https://aldaniel.narod.ru/strmbs.htm}
MONDAY BEGINS ON SATURDAY by Arkadi \& Boris Strugatski.
Translated by Leonid Renen.
DAW BOOKS, INC. DONALD A. WOLLHEIM, PUBLISHER
1301 Avenue of the Americas,  New York, N. Y. 10019
ENGLISH TRANSLATION copyright © 1977
{\em .. It's nonsense to look for  a solution if it already exists.  We are talking about
how to  deal with a  problem that has no solution. ..}

\bibitem{erofeev} 
\url{https://books.google.co.jp/books?id=9B7aarMNw4AC&q=opera#v=snippet&q=opera&f=false}
Venedikt Erofeev. Moscow to the end of the line. (Moskva-Petushki)
Publisher:	New York : Taplinger Pub. Co., 1980, p.80: .. Get up.
Go wash yourself and finish your divine opera {\em Khovanshina!}.

\bibitem{stru}
\url{https://www.rusf.ru/abs/books/sot-201.htm}\\
А.Н.Стругацикй, Б.Н.Стругацкий. Сказка о тройке - 2. 1968 {\em Лифт должен выдерживать прямое попадание в кабину самого необученного академика.} 
In Russian. English translation:
\url{https://royallib.com/read/Strugatsky_Arkady/Tale_of_the_Troika.html}  Tale of the Troika by Arkady and Boris Strugatsky. PROLOG.
{\em "..Our slogan is ‘elevators for everyone.’ No matter who. The elevator must be able to withstand the entrance of the least-educated academician.”}

\bibitem{ecale}
\url{https://mathdoc.emath.fr/PMO/PDF/E_ECALLE_67_74_09.pdf}
Jean Ecalle. Teorie des Invariants Holomorphes.  Universite Paris XI U.E.R. Mathematique 91405 ORSAY France (1974) %cite17

\bibitem{IRENE}
\url{https://people.math.sfu.ca/~cbm/aands/page_803.htm}
Abramowitz and Stegun: Handbook of Mathematical Functions. 
Author	Milton Abramowitz and Irene Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
Publisher	United States Department of Commerce, National Bureau of Standards (NBS), 1964. p.803: Zeta function.

\bibitem{pomeau}\url{https://link.springer.com/article/10.1007/BF01197757#page-1} ~ 
Y.Pomeau, P.Manneville. Intermittent Transition to Turbulence in Dissipative Dynamical Systems.
Communications in Mathematical Physics, 1980, v.74, p.189.%cite19

\bibitem{Eckmann81}
\url{https://users-phys.au.dk/fogedby/chaos/Eckmann81.pdf}\\
\url{https://rmp.aps.org/abstract/RMP/v53/i4/p643_1}\\
J.-P. Eckmann. Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654 (1981). %cite20

\bibitem{L3}%cite21
\url{https://www.sciencedirect.com/science/article/pii/003040188590330X}
{%\em 
Hnilo A. A.} Chaotic (as the logistic map) laser cavity. Optics Communications {\bf 53}. (1985). P.194-196. %cite21

\bibitem{L2} %cite22
\url{https://www.sciencedirect.com/science/article/pii/S1049250X06540108}
{%\em
Illing L., Gauthier D.J., Roy R.}
Controlling optical chaos, spatio-temporal dynamics, and patterns.
Advances in atomic, molecular and optical physics. {\bf 54}, (1987). P. 616-797. %cite22

\bibitem{klavanur} 
\url{https://mizugadro.mydns.jp/t/index.php/Klava_and_Nura}  (1989; English translation 2016)

\bibitem{walker} %cite24
\url{https://www.sciencedirect.com/science/article/pii/0022247X9190029Y}
Peter L Walker, On the solutions of an abelian functional equation, Journal of mathematical
analysis and applications 155 (1991), 93–110.

\bibitem{L1992}%cite25
{\em Tufillaro N., Tyler A., Jeremiah R.} 
An experimental approach to nonlinear dynamics and chaos. 1992. NY. 

\bibitem{walter} \url{https://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf}
W.Bergweiler. Iteration of meromorphic functions. Bulletin (New Series) of the American Mathematical society, v.29, No.2 (1993) p.151-188.%cite26

\bibitem{shapiro} %cite27
\url{https://www.math.msu.edu/~shapiro/Pubvit/Downloads/RieszExpo/RieszExpo.pdf}
Joel H. Shapiro. Composition Operators and Schr\"oder’s Functional Equation. Contemporary Mathematics 213 (1998), 213–228.

\bibitem{edua}
\url{https://elibrary.ru/item.asp?id=14996367} \\
\url{https://mizugadro.mydns.jp/PAPERS/gomez.pdf}
D.Kouznetsov, E.Gomez. 
Self-Modulation of Optical Pulses in a Kerr Medium and Limits of the Single-Mode Approximation
Optics and Spectroscopy, 1999, v.87, No.4, pp.594-602.

\bibitem{L2000}%cite28
{\em Strogatz S.} Nonlinear Dynamics and Chaos. NY, 2000. 

\bibitem{keller1} \url{https://edit.ethz.ch/ulp/publications/paper/2001/152__APB_73__653__2001_.pdf}\\
R.Paschotta, U.Keller. Passive mode locking with slow saturable absorbers. Appl. Phys. B 73, 653–662 (2001) %cite29

\bibitem{L2003} %cite30
{\em Sprott J. C.} Chaos and Time-Series Analysis. Oxford 2003. %cite30

\bibitem{slab1} %cite31
\url{https://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1278604}
\url{https://mizugadro.mydns.jp/PAPERS/ieee3.pdf}
D.Kouznetsov, J.V.Moloney. Highly efficient, high gain, short wavelenght and power scalable incoherent diode slab-pumped fibre laser amplifier. -- IEEE Journal of Quantum Electronics, v.39, No.11, p.1452-1461 (2003).

\bibitem{slab2} %cite32
\url{https://mizugadro.mydns.jp/PAPERS/01278604.pdf} ~
D.Kouznetsov, J.V.Moloney. Slab delivery of incoherent pump light to double-clad fibre amplifiers: An analytic approach. -- IEEE Journal of Quantum Electronics, 2004, v.40, Issue 4, p.378-383.

\bibitem{slab3}  %cite33
\url{https://mizugadro.mydns.jp/PAPERS/ieee3.pdf} \\
P.Kano, D.Kouznetsov, J.V.Moloney, M.Brio. Slab delivery of incoherent pump light to double-clad fibre amplifiers: Numerical simulations. -- IEEE Journal of Quantum Electronics, 2004, v.40, No.9, p.1301-1305.

\bibitem{keller2} \url{https://www.ulp.ethz.ch/publications/paper/2004/200__APB_79__331__2004_.pdf}
M.Haiml, R.Grange, U.Keller. Optical characterization of semicondutcor saturable absorbers. Appl.Phys. B 79, 331-339 (2004)  %cite34

\bibitem{kandi}
\url{https://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf} В.М.Гордиенко, В.К.Новик. О времени и факультете, о кафедре и о себе... 70-ти летие профессора В.П.Кандидова. (2007.12.05, In Russian). 
\footnote{
{\em По итогам студенческого голосования победителями оказались значок с изображением рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде корня из факториала, вписанными в букву Ф. Этот значок, созданный студентом кафедры биофизики А.Сарвазяном, привлекал своей простотой и выразительностью. Тогда эмблема этого значка подверглась жесткой критике со стороны руководства факультета, поскольку она не имеет физического смысла, математически абсурдна и идеологически бессодержательна. Уступая просьбам комсомольцев, партком (профессор А.И.Кузовников) согласился на то, чтобы значок выпустить малой серией, только на один год, с тем, чтобы в дальнейшем создать настоящий символ для студентов факультета. Сейчас, как не парадоксально, эмблема первого значка вошла в официальный бланк и печать факультета.}\\
English translation:\\
{\em According to the results of the student vote, winners were the icon with the image of the lever, raising the Earth, and the current from the well-known emblem of the root of factorial, inscribed in the letter $\Phi$. This icon is created by the student  Department of Biophysics, A.Sarvazyanom; it is attracted by its simplicity and expressiveness. Then the icon emblem has undergone harsh criticism from the Faculty of management, since it has no physical meaning, mathematically absurd and ideologically vacuous. Yielding to the requests of the Komsomol, the party committee (Professor A.I.Kuzovnikov) agreed that the icon to release a series of small, only one year, so that in the future to create a real symbol for the Faculty students. Now, paradoxically, the emblem of the first icon was included in the official form and faculty printing. ..}
}

\bibitem{sado}
\url{https://www.nkj.ru/archive/articles/1023/} В.Садовничий. ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250! Наука и жизнь, 2005, номер 1. (In Russian)

\bibitem{akira0} %cite35
\url{https://www.ils.uec.ac.jp/~dima/PAPERS/CoupledOptRev.pdf} 
\url{https://mizugadro.mydns.jp/PAPERS/CoupledOptRev.pdf}\\
D.Kouznetsov, J.-F.Bisson, A.Shirakawa, K.Ueda. Limits of coherent addition of lasers: simple estimate. -- Optical Review, 2005, v.12, No.6 (2005) 445-447.

\bibitem{thin}  %cite36
\url{https://mizugadro.mydns.jp/PAPERS/2005josab.pdf} %article;
D.Kouznetsov, J.-F.Bisson, K.Takaichi, K.Ueda. High-power single mode solid state laser with short unstable cavity. JOSA B, v.22, Issue 8, p.1605-1619 (2005).
\bibitem{thine}
\url{https://mizugadro.mydns.jp/PAPERS/2005josaber.pdf} Ibid: Erratum.  %cite37

\bibitem{fidi2} \url{https://www.nature.com/nphoton/journal/vsample/nsample/full/nphoton.2006.6.html}
Hamamatsu Photonics K.K. Laser group. The fibre Disk Laser explained.
Nature Photonics sample, - pp14 - 15 (2006).% doi:10.1038/nphoton.2006.6
 %cite38
 
\bibitem{perpe1}
\url{https://apl.aip.org/resource/1/applab/v88/i22/p221117_s1} Wenxue Li, Haifeng Pan, Liang’en Ding, Heping Zeng, Wei Lu, Guangjun Zhao, Chengfeng Yan, Liangbi Su, and Jun Xu. Efficient diode-pumped $\rm Yb:Gd_2 Si O_5$ laser. Appl.Phys.Lett. v.88, 221117 (2006)

\bibitem{perpe2}
\url{https://dx.doi.org/10.1364/OE.14.006681} Wenxue Li, Shixiang Xu, Haifeng Pan, Liangen Ding, Heping Zeng, Wei Lu, Chunlei Guo, Guangjun Zhao, Chengfeng Yan, Liangbi Su, and Jun Xu. Efficient tunable diode-pumped Yb:LYSO laser. Optics Express, Vol. 14, Issue 15, pp. 6681-6686 (2006), doi 10.1364/OPEX.14.000686

\bibitem{perpe3}%43
\url{https://www.sciencedirect.com/science/article/pii/S0038109805010094}\\
\url{https://210.72.9.5:8080/thesis/2006/2/17/142414.pdf} C.Yan, G.Zhao, L.Zhang, J.Xu, X.Liang, D.Juan, W.Li, H.Pan, L.Ding, H.Zeng. A new Yb-doped oxyorthosilicate laser crystal: $\rm Yb: Gd_2 Si O_5$. Solid State Comm. v.137, 451-455 (2006)

\bibitem{perpe4}%44
\url{https://www.ils.uec.ac.jp/~dima/PAPERS/ApplPhysLett_90_066101.pdf} \\
\url{https://mizugadro.mydns.jp/PAPERS/ApplPhysLett_90_066101.pdf} D. Kouznetsov. Comment on Efficient diode-pumped Yb:Gd2SiO5 laser. Appl.Phys.Lett.88,221117(2006). %APL, v.90, p.066101 (2007)

\bibitem{perpe5}%45
\url{https://mizugadro.mydns.jp/PAPERS/2007mc.pdf} D.Kouznetsov. Broadband laser materials and the McCumber relation. Chinese Optics Letters v.5, p.S240–S242 (2007)

\bibitem{perpe6}%46
\url{https://apl.aip.org/resource/1/applab/v90/i6/p066103_s1} G.Zhao. Response to Comment on Efficient diode-pumped Yb:Gd2SiO5 laser. Appl. Phys. Lett. v.90, 066101 (2007) %APL 90 p.066103 (2007)

\bibitem{BissonEffect}%47
\sx{.96}{\url{https://mizugadro.mydns.jp/PAPERS/ApplPhysLett_90_201901.pdf}}
J.-F.Bisson, D.Kouznetsov, K.Ueda, S.T.Fredrich-Thornton, K.Petermann, G.Huber. Switching of emissivity and photoconductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics. -- Applied Physics Letters, 90, 201901 (2007) ; doi:10.1063/1.2739318

\bibitem{hooshmand}  %48
\url{https://www.tandfonline.com/doi/full/10.1080/10652460500422247} 
M.H.Hooshmand. Ultra power and ultra exponential functions. Integral Transforms and Special Functions, 2006, 17 (8): 549–558. %doi:10.1080/10652460500422247.

\bibitem{bisson}  %49
\sx{.98}{\url{https://apl.aip.org/resource/1/applab/v90/i20/p201901_s1}}
\sx{.92}{\url{https://mizugadro.mydns.jp/PAPERS/ApplPhysLett_90_201901.pdf}}
J.-F.Bisson, D.Kouznetsov, K.Ueda, T.Fredrich-Thornton, K.Petermann, G.Huber. Switching of emissivity and photoconductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics. -- Applied Physics Letters, 90, 201901 (2007).%; doi:10.1063/1.2739318

\bibitem{tea} %50
\url{https://mizugadro.mydns.jp/t/index.php/Maple_and_tea}
\url{https://budclub.ru/k/kuznecow_d_j/maple.shtml}\\
\url{https://en.wikisource.org/wiki/Maple_and_Tea}\\
D.Kouznetsov. Maple and Tea. (2007)

\bibitem{kandidov} \url{https://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf}\\
В.М.Гордиенко, В.К.Новик. О времени и факультете, о кафедре и о себе...
70-ти летие профессора В.П.Кандидова. (2007.12.05, in Russian) %51

\bibitem{gac} %52
\url{https://inis.iaea.org/search/search.aspx?orig_q=RN:39069239}
J.M.Gac, J.J.Zebrowski. Giant suppression of the activation rate in dynamical systems with chaotic transitions.
Acta Physica Polonica B, 2008, v.39, p.1019. 

\bibitem{analuxp} %53
\url{https://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html}\\
\url{https://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf}\\
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation {\bf 78} (2009), 1647-1670.

\bibitem{L1} %54 WHY \sx{\url{..}} fails below??
{\url{https://link.springer.com/article/10.3103/S1068335609050054#page-1}}
{Gorbuklov M.V., Maslova Yu.Ya., Chaban O.I., Shabalin Yu.V.}
Period doubling cascade and deterministic chaos in a laser self-mode-locked by the combination of intertial negative and positive feedbacks.
Bulletin of the Lebedev Physics Institute, {\bf 36}. No.5. (2009). P.150-156.   

\bibitem{L2009} %cite46
\url{https://www.springerlink.com/content/d73u677140l3j826/}\\
{Bjerkl\"ov K.}
SNA’s in the Quasi-Periodic Quadratic Family
Communications in Mathematical Physics. {\bf 286}. 2009. P.137--161
\JP{
SNA’s in the Quasi-Periodic Quadratic Family
Journal Communications in Mathematical Physics
Publisher       Springer Berlin / Heidelberg
ISSN    0010-3616 (Print) 1432-0916 (Online)
Issue   Volume 286, Number 1 / February, 2009
DOI     10.1007/s00220-008-0626-y
Pages   137-161
Subject Collection      Physics and Astronomy
SpringerLink Date       Tuesday, September 09, 2008
}%

\bibitem{topology} \sx{.9}{\url{https://mizugadro.mydns.jp/t/index.php/Lesson_of_quantum_topology}}

\bibitem{elutin} %57
P.V.Elutin. The holomorphic extension of the logistic mapping is ambitious program...
The continual generalization of the logistic mapping will worth your time and efforts, if you can reveal the model
that gives the law of the evolution $F(t)$ - or determine the characteristics $F(t)$, boundaries of the time derivatives. - Private communication, Oct 9, 2009 5:32 PM.
({\em Елютин П. В. Думаю, задача, понятая как ``КОНТИНУАЛЬНОЕ ОБОБЩЕНИЕ ЛОГИСТИЧЕСКОГО ОТОБРАЖЕНИЯ'', будет стоить Ваших усилий и времени, если удастся выявить модель, порождением которой является закон движения $X(t)$ - или определить континуальные характеристики $X(t)$ (например, границы значений производных по времени})
%Частное сообщение, Oct 9, 2009 5:32 PM.

\bibitem{physmat} %58
\url{https://iopscience.iop.org/1742-6596/237/1/012016/pdf/1742-6596_237_1_012016.pdf}\\
P.G.L.Leach and M.C.Nucci. Point and counterpoint between Mathematical Physics and Physical Mathematics.
Journal of Physics: Conference Series {\bf 237} (2010) 012016.

\bibitem{euclid} \url{https://projecteuclid.org/euclid.jpm} The Journal of Physical Mathematics is a scholarly, peer-reviewed, and fully refereed journal dedicated to publishing high-quality research articles and selected review articles in the fields of physically motivated mathematics and mathematical methods of physics. //
The Journal of Physical Mathematics was published by Ashdin Publishing until 2013. Beginning in 2017, this journal will no longer be included in Euclid Prime or distributed by Project Euclid. 

\bibitem{sqrt2} %cite60
\url{https://www.jointmathematicsmeetings.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/S0025-5718-10-02342-2.pdf}
\url{https://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html} 
\url{https://eretrandre.org/rb/files/Kouznetsov2009_215.pdf}\\
\url{https://www.ils.uec.ac.jp/~dima/PAPERS/2010q2.pdf}\\
\url{https://mizugadro.mydns.jp/PAPERS/2010q2.pdf}\\
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

\bibitem{facit} %cite51
\url{https://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1} \\
\url{https://mizugadro.mydns.jp/PAPERS/2010superfae.pdf}
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12

\bibitem{vladir}  %cite52
\url{https://www.emis.ams.org/journals/VMJ/articles/2010_2_4.pdf}
\url{https://mizugadro.mydns.jp/PAPERS/2009vladir.pdf}  
\url{https://www.ils.uec.ac.jp/~dima/PAPERS/2009vladir.pdf}
Д.Кузнецов. Тетрация как специальная функция. Владикавказский математический журнал, 2010, т.12, вып. 2, стр.31-45. 
(In Russian. See the English version below.)

\bibitem{vladie}  %cite53
\url{https://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf}\\
\url{https://mizugadro.mydns.jp/PAPERS/2010vladie.pdf} ~
D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. 

\bibitem{factorial}%cite54
\url{https://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1}
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. 
(Russian version: Вестник Московского Университета, серия 3, 2010 No.1, стр.8-14.)

\bibitem{bitbucket} %65
\sx{.96}{https://cdn.bitbucket.org/bo198214/bunch/downloads/main.pdf} H.Trappmann, D.Kouznetsov. 5+ methods for real analytic tetration. June 28, 2010. (since 2020, does not open)

\bibitem{ufn}%cite66
\url{https://ufn.ru/tribune/trib120111.pdf} \\
\url{https://mizugadro.mydns.jp/t/index.php/Place_of_science_in_the_human_knowledge}
Д.Кузнецов. Место науки и физики в человеческом знании. УФН, т.181 ``Трибуна'', стр.1-9 (2010, In Russian. See the English version below).

\bibitem{mesto} %cite67
\url{https://mizugadro.mydns.jp/t/index.php/Mestoe}\\
D.Kouznetsov. Place of science in human knowledge. (2011-2020)

\bibitem{logistic}%78
\sx{.99}{\url{https://www.ils.uec.ac.jp/~dima/PAPERS/2010logistie.pdf}}
\url{https://mizugadro.mydns.jp/PAPERS/2010logistie.pdf}\\
\url{https://link.springer.com/article/10.3103/S0027134910020049}
D.Kouznetsov. Holomorphic extension of the Logistic sequence. Moscow State University Physics Bulletin, 
v.65, p.91-98 (2010).
% 3 (2010) No.2, стр.23-30. 
The logistic problem is formulated in terms of the Superfunction and Abelfunction of the quadratic transfer function 
H(z)=uz(1-z). The Superfunction F as holomorphic solution of equation H(F(z))=F(z+1) generalizes the logistic sequence to the complex values of the argument z. The efficient algorithm for the evaluation of function F and its inverse function, id est, the Abelfunction G are suggested; F(G(z))=z. The halfiteration h(z)=F(1/2+G(z)) is constructed; in wide range of values z, the relation h(h(z))=H(z) holds. For the special case u=4, the Superfunction F and the Abelfunction G are expressed in terms of elementary functions.

\bibitem{mizugadronumber} 
\url{http://budclub.ru/k/kuznecow_d_j/mizugade.shtml}
%\url  https://budclub.ru/k/kuznecow_d_j/mizugade.shtml} D.Kouznetsov. Mizugadro's number. (2010)\\
\url{https://mizugadro.mydns.jp/t/index.php/Mizugadro_number_dream}  D.Kouznetsov. Mizugadro's number. (2010-2017) 

\bibitem{yoneda} Hitoki Yoneda. Recovery of intensity inside the amplifier from its transfer function is impossible. Private communication, 2010. %cite59

\bibitem{kaminski} Aleksander Kaminskii. 
There is no sense to analyse the transfer function of the amplifier without to specify the model for the active medium. Private communication,
2011. %cite60

\bibitem{uniabel}%cite63
\url{https://link.springer.com/article/10.1007/s00010-010-0021-6}
\url{https://pdfs.semanticscholar.org/203a/d0586b625d2648aa6e22f36da7189f9041bd.pdf}
%\url{https://www.springerlink.com/content/u7327836m2850246/}\\
\url{https://www.ils.uec.ac.jp/~dima/PAPERS/2011uniabel.pdf}\\
\url{https://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf} \\
H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)

\bibitem{akira1} Akira Shirakawa. It is not possible to reconstruct the intensity inside the amplified just from its transfer function.
Private communication, 2011.%cite64

\bibitem{akira2} Akira Shirakawa. The formalism of superfunctions is too complicated and therefore cannot be used.
Private communication, 2012.10.09.%cite65

\bibitem{kur}%cite66
\url{https://web.abo.fi/fak/mnf/mate/kurser/dynsyst}\\
Kursernas Hemsidor. Introduction to Dynamical Systems. 2012.  
Derivation of Niclas Carlsson's formula. Let function $f(x)$ be $sin(x)$. 
We want to evaluate, approximately .the value of the n-th iterate of $f(x)$. 
Niclas Carlsson obtained empirically the following formula $f^n(x)\approx \sqrt{\frac{3}{n}}$, $x\approx 1$, $n$ large .. If the formula is correct it will take $3 \cdot\! 10^{10}$.. steps to reach 0.00001 starting from 1. ..After 60000 iterations the value is 0.0071 and the discrepancy  -4.7323e-007. 
 
\bibitem{car} 
\url{http://www.physics.miami.edu/~curtright/Schroeder.html}
%\url{https://www.physics.miami.edu/~curtright/Schroeder.html}
Thomas Curtright. Continuous iterates continue to be interesting, after 150 years of study. (2011-2013)
As a first illustration, we display the continuous iterates of the sine function, $\sin_{[t]}(x)$. Note that the maximum values at $x = \pi/2$ are approximately given by $\exp[(1-\sqrt{t}) \ln(\pi/2)]$.%cite67

\bibitem{fizmat} %Л. С. СЫЧЕВА.«ФИЗИЧЕСКАЯ МАТЕМАТИКА» АРХИМЕДА, ФОРМИРОВАНИЕ ИНТЕГРАЛЬНОГО ИСЧИСЛЕНИЯ И МЕХАНИЗМЫ НОВАЦИЙ В МАТЕМАТИКЕ. %ΣΧΟΛΗ Vol. 6. 2 (2012).
\url{https://www.nsu.ru/classics/schole/6/6-2-sych.pdf} 
Л.С.Сычева. Физическая математика Архимеда, формирование интегрального исчисления и механизмы новаций в математике. Школа, том 6, номер 2 (2012, in Russian).%cite68

\bibitem{e1e}%cite69
\url{https://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf}
\sx{.96}{\url{https://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf}}
\url{https://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf} ~
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation, 2012,
{\bf 81},
p.2207-2227.


\bibitem{kvashenko}%cite70
\url{http://mmmf.msu.ru/lect/kvashenko.html}
%\url{https://mmmf.msu.ru/lect/kvashenko.html}
Квашенко Андрей Николаевич. Драконы. (2007-2013, in Russian)

\bibitem{unichtozheniepr}%80
\url{https://mizugadro.mydns.jp/PAPERS/2013physToday.pdf}
D.Kouznetsov. Corruption in Russian science. Preprint, 2013.

\bibitem{unichtozhenie}%81
{\url{https://mizugadro.mydns.jp/t/index.php/Destruscir}}
Д.Кузнецов. Уничтожение науки в РФ. (Russian version)

\bibitem{2013jmp}%cite82
\sx{.88}{\url{https://www.scirp.org/journal/PaperInformation.aspx?PaperID=36560}}
\url{https://mizugadro.mydns.jp/PAPERS/2013jmp.pdf} \\ 
D.Kouznetsov. TORI axioms and the applications in physics. Journal of Modern Physics, 2013, v.4, p.1151-1156.

\bibitem{thailand}%cite83
\url{https://www.ingentaconnect.com/content/asp/asl/2013/00000019/00000003/art00071}\\
\url{https://mizugadro.mydns.jp/PAPERS/2012thaiSuper.pdf}\\
D.Kouznetsov. Recovery of Properties of a Material from Transfer Function of a Bulk Sample (Theory). Advanced Science Letters, Volume 19, Number 3, March 2013, pp. 1035-1038(4).

\bibitem{2013or}%\bibitem{superlaser}%cite74
\sx{.94}{\url{https://link.springer.com/article/10.1007/s10043-013-0058-6}}
\url{https://mizugadro.mydns.jp/PAPERS/2013orSuper.pdf}
%\url{https://mizugadro.mydns.jp/PAPERS/2013or2.pdf}
D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, V. 20, Issue 4, p. 321-326. 

\bibitem{log}%cite75
\url{https://arxiv.org/abs/1008.1409v2}
Henryk Trappmann. The Intuitive Logarithm. October 14, 2013.

\bibitem{guess}%cite76
Michael Moldenhauer. Question about tetration method. Private communication, Sat, 9 Nov 2013 01:11:53 -0800 (PST).
{\em ..how did you come up with the high-quality initial approximation fit\_3 mentioned in your paper about
the tetrational function? }

\bibitem{sutra}%cite77
\url{https://mizugadro.mydns.jp/PAPERS/2013hikari.pdf}\\
\url{https://pdfs.semanticscholar.org/6b95/32be1edcd5043e5661861ada523626d91b04.pdf}
%\url{https://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf}\\
%\url{https://mizugadro.mydns.jp/PAPERS/2013hikari.pdf}\\
D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541. 

\bibitem{pentation}
\url{http://www.sciencepublishinggroup.com/journal/paperinfo.aspx?journalid=147&doi=10.11648/j.acm.20140306.14}
%\url{http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=147&doi=10.11648/j.acm.20140306.14}
%\url{https://www.sciencepublishinggroup.com/journal/paperinfo?journalid=147&doi=10.11648/j.acm.20140306.14}
Dmitrii Kouznetsov. Evaluation of Holomorphic Ackermanns.
Applied and Computational Mathematics 
Volume 3, Issue 6, December 2014, Pages: 307-314 

\bibitem{bonet}
\url{https://dx.doi.org/10.1007/s00020-014-2175-4}
Jose Bonet, Pawel Domanski.
Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions.
Integral Equations and Operator Theory, 
Published online: 2014 August 19. 

\bibitem{susin}
\url{https://www.pphmj.com/references/8246.htm}\\
\url{https://mizugadro.mydns.jp/PAPERS/2014susin.pdf}
D.Kouznetsov. Super sin. Far East Jourmal of Mathematical Science, v.85, No.2, 2014, pages 219-238.

\bibitem{paulsen2016}
\url{https://kkms.org/index.php/kjm/article/view/428/302}
W.Paulsen.
Finding the natural solution to f(f(x))=exp(x).
The Korean Journal of Mathematics, 2016,  Vol 24, No 1 (2016), p. 81-106.

\bibitem{paulsen}
\url{http://myweb.astate.edu/wpaulsen/tetration.html}
%\url{https://myweb.astate.edu/wpaulsen/tetration.html}
William Paulsen. Tetration is repeated exponentiation. (2016-2020).
%{\em The authors also devised a quick way of computing F(z) to 50 places of accuracy. From this, they could have Mathematica draw complex contour maps of the function F(z). For example, here is the contour map for the base b = 2: ..}

\bibitem{paulsencowgill}
\url{https://link.springer.com/article/10.1007/s10444-017-9524-1}
William Paulsen, Samuel Cowgill. Solving $F(z \!+\! 1) = b^{F(z)}$ in the complex plane.
Advances in Computational Mathematics, v.43, p.1261-1282 (2017) %07 March 2017.

\bibitem{cowgill}
\url{https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1}
Samuel Cowgill. Exploring Tetration in the Complex Plane. 
Arkansas State University, May 2017. ProQuest number: 10263680.

\bibitem{paulsen2019}
\url{https://www.researchgate.net/publication/325532999_Tetration_for_complex_bases}
William Harold Paulsen. Tetration for complex bases.
%Article (PDF Available) in 
Advances in Computational Mathematics, v.45, p. 243–267(2019)
% · June 2018 wit
%h 201 Reads 
%DOI: 10.1007/s10444-018-9615-7
%Cite this publication

\bibitem{wikiabel} \url{https://en.wikipedia.org/wiki/Abel_equation}

\bibitem{nest} \url{https://reference.wolfram.com/mathematica/ref/Nest.html}

\bibitem{conto.cin} \url{https://mizugadro.mydns.jp/t/index.php/Conto.cin}

\bibitem{cz.conto.cin} \url{https://en.citizendium.org/wiki/Conto.cin}


\bibitem{science} \url{https://mizugadro.mydns.jp/t/index.php/Science}

\bibitem{sciencer} \url{https://mizugadro.mydns.jp/t/index.php/Hayka}
Наука

\bibitem{Hayka} \url{https://budclub.ru/k/kuznecow_d_j/hayka.shtml} Наука

\bibitem{Cauchy} \sx{.98}{\url{https://en.wikipedia.org/wiki/Cauchy's_integral_formula}}

\bibitem{toritet} \url{https://mizugadro.mydns.jp/t/index.php/Tetration}

\bibitem{cztet} \url{https://en.citizendium.org/wiki/Tetration}

\bibitem{tania} \url{https://mizugadro.mydns.jp/t/index.php/Tania_function} \\
D.Kouznetsov. Tania function. 

\bibitem{doya} \url{https://mizugadro.mydns.jp/t/index.php/Doya_function} \\
\url{https://www.ils.uec.ac.jp/~dima/e/Doya} ~

\bibitem{Keller} \url{https://mizugadro.mydns.jp/t/index.php/Keller_function}\\
\url{https://www.ils.uec.ac.jp/~dima/e/Keller} ~

\bibitem{Shoka}
\url{https://mizugadro.mydns.jp/t/index.php/Shoka_function}\\ 
\url{https://www.ils.uec.ac.jp/~dima/e/Shoka} ~

\bibitem{collatz} \url{https://mizugadro.mydns.jp/t/index.php/Holomorphic_extension_of_the_Collatz_subsequence}

\bibitem{LambertW} 
\url{https://mizugadro.mydns.jp/t/index.php/LambertW}

\bibitem{LambertWm} \url{https://mathworld.wolfram.com/LambertW-Function.html}

\bibitem{LambertWs} \url{https://en.citizendium.org/wiki/Lambert_W_function}

\bibitem{LambertWw} \url{https://en.wikipedia.org/wiki/Lambert_W_function}

\bibitem{david} \url{https://www.researchgate.net/publication/228724354_The_Wright_o_Function}
%    https://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf}\\
Robert M. Corless, David J. Jeffrey. The Wright $\omega$ function. Artificial intelligence, January 2002.
%DOI: \doi{10.1007/3-540-45470-5_10}

\bibitem{valerie}
\url{http://valerie.doya.free.fr/Doya_CV.html}
%  https://valerie.doya.free.fr/Doya_CV.html}
Valérie DOYA. Laboratoire de Physique de la Matière Condensée, Parc Valrose, 06108 Nice Cedex2, France. (2014)

\bibitem{bookworm}
\url{https://commons.wikimedia.org/wiki/File:Carl_Spitzweg_021.jpg}
Carl Spitzweg. The Bookworm. 1850.
\end{thebibliography}

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Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected.
For a given transfer function T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfunction is inverse of F.
In particular, superfunctions of factorial, exponent, sin are suggested. Also, the holomorphic extensions of the logistic sequence and those of the Ackermann functions are considered. 
Among ackermanns, the tetration (mainly to the base b>1) and natural pentation (to base b=e) are presented. The efficient algorithm for the evaluation of superfunctions and abelfunctions are described. The graphics and complex maps are plotted. The possible applications are discussed.
Superfunctions significantly extend the set of functions that can be used in scientific research and technical design.
Generators of figures are loaded to the site TORI, https://mizugadro.mydns.jp for the free downloading. With these generators, the Readers can reproduce (and modify) the figures from the Book. The Book is intended to be applied and popular. I try to avoid the complicated formulas, but some basic knowledge of the complex arithmetics, Cauchy integral and the principles of the asymptotical analysis should help at the reading.
}\end{parbox}}}

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\vskip 4pt
Graduated from the Physics Department of the
Moscow State University (1980). Work: USSR, Mexico, USA, Japan.
Century 20: Proven the\\
quantum stability of the optical soliton, suggested\\
the low bound of the quantum noise of nonlinear\\
amplifier, indicated the limit of the single mode\\
approximation in the quantum optics. Century 21:\\
Theory of ridged atomic mirrors, formalism of\\
superfunctions, TORI axioms. 

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