# Superfunctions

Front cover of the Book
back cover of the Book
Example of the explicit plot from the Book: plot of tetration, $y=\mathrm{tet}_b(x)$ versus $x$ for various $b\!>\!1$, Figure 17.1
Example of the complex map from the Book: map of the truncated Taylor expansion of abelsine AuSin, Figure 22.2
Literature cited in the Book, Figure 21.1
Literature not cited in the Book, Figure 21.3

Superfunctions is book, English version of the Russian book Суперфункции, that is in preparation since year 2015.

The English version is unlikely to be ready in year 2015 (as it is marked in the project of the cover at right), but there are still some hopes for year 2018.

## Covers

At the front cover, the complex map of natural tetration is shown.

The following topics are indicated: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics and complex maps.

The back cover suggests short abstract of the Book and few notes about the Author.

Assume some given holomorphic function $T$. The superfunction is holomorphic solution F of equation $T(F(z))=F(z+1)$

The Abel function (or abelfunction) is the inverse of superfunction, $G=F^{-1}$

The abelfunction is solution of the Abel equation $G(T(z))=G(z)+1$

As the superfunction $F$ and the abelfunction $G=F^{-1}$ are established, the $n$th iterate of transfer function $T$ can be expressed as follows: $T^n(z)=F(n+G(z))$

This expression allows to evaluate the non-integer iterates. The number n of iterate can be real or even complex. In particular, for integer $n$, the iterates have the common meaning: $T^{-1}$ is inverse function of $T$,
$T^0(z)=z$,
$T^1(z)=T(z)$,
$T^2(z)=T(T(z))$,
$T^3(z)=T(T(T(z)))$,
and so on. The group property holds: $T^m(T^n(z))=T^{m+n}(z)$

The book is about evaluation of the superfunction $F$, the abelfunction $G$ and the non-integer iterates of various transfer functions $T$.

The special notation is used in through the book; the number of iterate is indicated as superscript. For example, In these notations, $\sin^2(z)=\sin(\sin(z))$, but never $\sin(z)^2$.
This notation is borrowed from the Quantum mechanics, where $P^2(\psi)=P(P(\psi))$, but never $P(\psi)^2$.

Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. For a giver transfer function T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfuction is inverse of F. In particular, thesuperfunctions of factorial, exponent, sin; the holomorphic extensions of the logistic sequence and of the Ackermann functions are suggested. from ackermanns, the tetration (mainly to the base b>1) and pentation (to base 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, http://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, Cauchi integral and the principles of the asymptotical analysis should help at the reading.

Dmitrii Kouznetsov

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: Theorem about boundary behaviour of modes of Dirichlet laplacian, Theory of ridged atomic mirrors, formalism of superfunctions, TORI axioms.

## Summary

The summary suggests main notations used in the Book:

$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

$\displaystyle F(z)=\frac{1}{2\pi \mathrm i} \oint \frac{F(t) \, \mathrm d t}{t-z}$ $~ ~ ~$ Cauchi integral

$\mathrm{tet}_b(z\!+\!1)=b^{\mathrm{tet}_b(z)}$ $~ ~ ~$ 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 $~$ $z\!\mapsto\! b^z$

$\displaystyle \mathrm{Tania}^{\prime}(z)=\frac{\mathrm{Tania}(z)}{\mathrm{Tania}(z)\!+\!1}$ $~ ~$ Tania function,$~$ $\mathrm{Tania}(0)\!=\!1$

$\displaystyle \mathrm{Doya}(z)=\mathrm{Tania}\big(1\!+\!\mathrm{ArcTania}(z)\big)$ $~ ~$ Doya function

$\displaystyle \mathrm{Shoka}(z)=z+\ln(\mathrm e^{-z}\!+\!\mathrm e \!-\! 1)$ $~$ Shoka function

$\displaystyle \mathrm{Keller}(z)=\mathrm{Shoka}\big(1\!+\!\mathrm{ArcShoka}(z)\big)$ $~ ~$ Keller function

$\displaystyle \mathrm{tra}(z)=z+\exp(z)$ $~ ~ ~$ Trappmann function

$\displaystyle \mathrm{zex}(z)=z\,\exp(z)$ $~ ~ ~ ~$ Zex function

$\displaystyle \mathrm{Nem}_q(z)=z+z^3+qz^4$ $~ ~ ~ ~$ Nemtsov function

## Disgraces

One go goals of the Book is to attract to Mathematical Analysis those readers, who easily confuse integral and logarithm. For these readers, several disgraces are added in the Book. Some go them are copipasted below. ?

### Krivchenkov

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:

- Could you explain me, please, why operation TO CLEAN-UP does not commute with operation TO DIRT-DOWN  ?

The student was content, she said:

- Thank you, Professor! This is very good example! Now I understand all the Quantum Mechanics!

### Evenings at a Farmhouse near Dikanka

I type this Prologue for those, who will not read this Book:

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.

For the experienced critics}, who read only two or three pages, in order to write the review:
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!
..

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 everyeye. . . . Nooneishisequalinpower—heisaGod! Butquite 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 ac- cepted 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 re- searcher is left by the roadside like the traveller without a family. Hard is his lot and bitterly he feels his loneliness.

### Fiber amplifier

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.

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 $T^z$.

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===Yukawa and Simpson===

If, at your presentation, at the first desk you see some 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 will understand at least the main idea of your talk.

### Black sheep

Figure 15.6

Two mathematicians go to the First International Congress on superfunctions. They discuss the color of the sheep they see from the train:

– Your assumption, dear colleague, is not obvious; it is not supported with observations. Yet, all what we can conclude, is, that there is at least one sheep in this country, and at least the right hand side of this sheep is black.

To explain this idea, I have drown this situation in the picture at right. I wanted to believe, that my drawings are not worse than those by Antoine de Saint-Exupéry.

### Linear and nonlinear optics

Figure 20.4

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, 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. 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.

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 coplex maps presented in this section. Then, these Readers can be qualified with specification “understand everything”. As those graduated from Fizfak.

Оne-legged friend of one Taylor asked him to sew the special pants with one leg. 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 dogs. 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.

### History

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 Historician came to the Emperor and gave him the pamphlet "A Brief History of the Imperial Family." The Emperor was old and ill, and could not even read that brochure. He asked Historian whether the brochure can be reduced. The Historian answered:

– No new edition is necessary. I’ll tell you right now: People were born, suffered and died.

## References

1. http://www.ils.uec.ac.jp/!dima/BOOK/437.pdf (a little bit out of date)

The book combines the main results from the following publications:

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation 78 (2009), 1647-1670.

http://www.jointmathematicsmeetings.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/S0025-5718-10-02342-2.pdf
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
http://eretrandre.org/rb/files/Kouznetsov2009_215.pdf
http://www.ils.uec.ac.jp/~dima/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.

D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.

D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12

http://mizugadro.mydns.jp/t/index.php/Place_of_science_in_the_human_knowledge D.Kouznetsov. Place of science and physics in human knowledge. English translation from http://ufn.ru/tribune/trib120111 Russian Physics:Uspekhi, v.191, Tribune, p.1-9 (2010)

http://www.ils.uec.ac.jp/~dima/PAPERS/2010logistie.pdf http://mizugadro.mydns.jp/PAPERS/2010logistie.pdf D.Kouznetsov. Continual generalisation of the Logistic sequence. Moscow State University Physics Bulletin, 3 (2010) No.2, стр.23-30.

http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
http://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf
H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation, 2012, 81, February 8. p.2207-2227.

http://www.ils.uec.ac.jp/~dima/PAPERS/2012or.pdf
Dmitrii Kouznetsov. Superfunctions for optical amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.

http://www.scirp.org/journal/PaperInformation.aspx?PaperID=36560