Difference between revisions of "Superfunction"

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{{top}}
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[[Superfunction]] comes from iteration of some function.
  
  +
For some holomorphic function \(T\) (called «[[Transfer function]]»), the [[Superfunction]] \(F\)
  +
is solution of the [[transfer equation]]
   
  +
\[
−
For some function <math>T</math> (which is called [[Transfer function]]) and for some constant <math>t</math>, the superfunction \(F\) could be defined with expression
  
  +
F(z\!+\!1) = T(F(z))
  +
\]
   
  +
Formalism of [[superfunction]]s is described in special book «[[Superfunctions]]» (2020)
−
\(\displaystyle {{F(z)} \atop \,} {= \atop \,}
 
−
{T^z(t) \atop \,} {= \atop \,}
  
−
{{\underbrace{T\Big(T\big(... T(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}T\!
  
−
\!\!\!\!\!}}\)
  
−  
−
then <math>F</math> can be interpreted as superfunction of function <math>T</math>.
  
−
Such definition is valid only for positive integer <math>z</math>.
 
−
The most research and appllications around the superfunctions are related with various extensions of superfunction; and analysis of the existence, uniqueness and ways of the evaluation.
  
−
<!-- In particular, :<math>S(1)=f(t)</math> !-->
  
−
For simple function <math>T</math>, such as addition of a constant or multiplication by a constant,
  
−
the superfunction can be expressed in terms of elementary function.
  
−
<!--
  
−
Historically, first non-elementary superfunction considered was super-exponential or [[tetration]], that corresponds to
 
−
!-->
 
−
In particular, the [[Ackernann functions]] and [[tetration]] can be interpreted in terms of [[superfunction]]s.
  
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<div style="float:right; width:180px">
  
−
<div style="width:190px">
  
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[[File:HellmuthKneserPhotoNormal.jpeg|190px]]<small>
  
−
[[Hellmuth Kneser]], 1958
  
 
<ref>
  
<ref>
  +
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics – 2020/7/28
  +
</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
  +
</ref>.<br>
  +
There is also Russian version «[[Суперфункции]]» (2014)
  +
<ref>
  +
https://mizugadro.mydns.jp/BOOK/202.pdf
  +
Дмитрий Кузнецов. Суперфункции. [[Lambert Academic Publishing, 2014]]
  +
</ref>.
  +
  +
[[Superfunction]]s allow to constrict non-integer iterates of holomorphic functions;
  +
in particular,
  +
the [[square root of factorial]] \(\ \phi\!=\!\sqrt{\ !\ }\ \)
  +
as solution <ref name="f">
  +
https://link.springer.com/article/10.3103/S0027134910010029 <br>
  +
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. (Russian version: p.8-14)
  +
</ref> of equation
  +
<div style="float:right; margin:0px 0px 0px 0px">
  +
{{pic|QFactorialQexp.jpg|360px}}<small><center>[[Complex map]]s \(\ p\!+\!\mathrm iq=\phi(z)\ \) and \(\ p\!+\!\mathrm iq\!=\!\varphi(z)\ \) <ref name="f"/>
  +
</center></small>
  +
</div>
  +
  +
\[
  +
\phi(\phi(z))=z!
  +
\]
  +
and
  +
the «[[square root of exponential]]» <ref name="k">
  +
http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi
  +
[[Hellmuth Kneser]]. Reelle analytische Lösungen der Gleichung \( \varphi(\varphi(x))=\mathrm e^x \) und verwandter Funktionalgleichungen.
  +
Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67
  +
</ref>
  +
\(\ \varphi\!=\!\sqrt{\exp}\ \) as solution of equation
  +
\[
  +
\varphi(\varphi(z))=\exp(z)
  +
\]
  +
  +
Maps of functions \(\phi\) and \(\varphi\) are shown in first figure.
  +
  +
<div class ="thumb tright" style="float:right; margin:72px 0px 0px 0px"; background-color:#fff>
  +
<p style="margin:40px -20px 30px 0px">&nbsp; {{pic|Sqrt(factorial)LOGOintegralLOGO.jpg|300px}}
  +
<p style="margin:-244px 0px -4px 164px">&nbsp; {{pic|HellmuthKneserPhotoNormal.jpeg|144px}}</p>
  +
<small><center>
  +
LOGO \(\sqrt{\ !\ }\) and [[Hellmuth Kneser|H.Kneser]] (\(\sqrt{\exp}\)) <ref name="f"/><ref>
 
https://opc.mfo.de/detail?photo_id=7607 On the Photo: Kneser, Hellmuth Location: Oberwolfach Author: Danzer, Ludwig (photos provided by Danzer, Ludwig) Source: L. Danzer, Dortmund Year: 1958 Copyright: L. Danzer, Dortmund Photo ID: 7607
  
https://opc.mfo.de/detail?photo_id=7607 On the Photo: Kneser, Hellmuth Location: Oberwolfach Author: Danzer, Ludwig (photos provided by Danzer, Ludwig) Source: L. Danzer, Dortmund Year: 1958 Copyright: L. Danzer, Dortmund Photo ID: 7607
−
</ref></small>
 +
</ref>
−
</div></div>
 +
</center></small>
  +
</div>
  +
  +
For a given transfer function \(T\) and its superfunction \(F\),
  +
the inverses function \(G\!=\!F^{-1}\) is the [[Abelfunction]].
  +
While for the [[Transfer function]] \(T\),
  +
the [[Superfunction]] \(F\) and
  +
the [[Abelfunction]] \(G\) are established, the \(n\)th iterate
  +
of the [[Transfer function]] can be expressed as follows:
   
  +
\[
−
==History==
  
  +
T^n(z)=F(n+G(z))
−
<div style="float:left;width:70px">
  
  +
\]
−
<div style="float:right;width:240px">
  
−
[[Image:Sqrt(factorial)LOGOintegralLOGO.jpg|230px]]<small>
  
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logos of Phys. and Math. depts of [[Moscow State University|MSU]]</small>
  
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</div></div>
  
−
<div class="thumb tright" style="float:right;width:190px">
  
−
<div style="width:200px">
  
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[[File:QexpMapT400.jpg|200px]] \(u+\mathrm i v=\sqrt{\exp} (x+\mathrm i y)\)
  
−
[[File:QfacMapT500a.jpg|200px]] \(u+\mathrm i v=\sqrt{!}(x+\mathrm i y)\)
  
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</div></div>
  
−
Analysis of superfunctions came from the application to the evaluation of fractional iterations of functions.
  
−
 
−
[[Superfunction]]s and their inverse functions ([[Abel function]]s) allow evaluation of not only minus-first power of a function (inverse function), but also any real and even complex [[iteration]] of the function.
 
   
  +
In this expression, number \(n\) of iterate has no need to be integer; it can be real or even complex.
−
Historically, the first function of such kind, iteration half of the exponential, id est,
 
  +
−
<math>\sqrt{\exp}~</math>, is considered in 1950 by [[Hellmuth Kneser]]
  
  +
In such a way, with the [[superfunction]] and the [[Abelfunction]], the [[transfer function]]
−
<ref name="kneser">
  
  +
can be iterated arbitrary times. To century 21, this property seems to provide the main application of [[superfunction]]s.
−
http://tori.ils.uec.ac.jp/PAPERS/Relle.pdf
  
  +
==History==
−
[[Helmuth Kneser]]
  
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Reelle analytische L¨osungen der Gleichung <math>\varphi(\varphi(x)) = e^x </math> und verwandter Funktionalgleichungen
  
−
[[Journal fur die reine und angewandte Mathematik]] '''187''' (1950) 56-67
  
−
</ref>. That time, no algorithm had been suggested for evaluation of this function.
  
   
  +
Functions \(\sqrt{\ !\ }\) and \(\sqrt{\exp} \) are mentioned century 20
−
Since 1960, function <math>\sqrt{!~}~</math> is accepted as logo of the Physics department of the [[Moscow State University]], see the first picture of the figure at left. <br>
  
  +
as LOGO pf Physics Department of the Moscow State University
−
Until year 2007, this symbol is believed to have no mathematical sense
  
−
<ref name="kandidov">
 +
<ref name="kandidov">(not available: http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf )
 
V.P.Kandidov. About the time and myself. (In Russian) 2007.05.18.
  
V.P.Kandidov. About the time and myself. (In Russian) 2007.05.18.
−
http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf:
  
 
.. По итогам студенческого голосования победителями оказались значок с изображением
  
.. По итогам студенческого голосования победителями оказались значок с изображением
 
рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде
  
рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде
Line 69: Line 91:
 
.. На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее. ..</ref><ref name="logo">Logo of the Physics Department of the Moscow State University. (In Russian);
  
.. На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее. ..</ref><ref name="logo">Logo of the Physics Department of the Moscow State University. (In Russian);
 
http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml
  
http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml
−
</ref>.
 +
</ref>
  +
and as holomorphic solution of \(\varphi(\varphi(z))=\exp(z)\) <ref name="f"/> suggested by [[Hellmith Kneser]],
  +
see pictures above.
   
  +
In century 20, before construction of formalism of [[superfunction]]s, these functions were not implemented; the [[complex maps]] of these functions were not plotted, although many publications about non-integer iterates of functions are observed.
−
The [[complex map]]s of functions \( \sqrt{\exp} \) and \(\sqrt{!} \) are shown in figures at right.
 
  +
The first robust implementations of non-trivial superfunctions and ablefunctions
  +
appear in years 2009-2010
  +
<ref name="f"/>
   
  +
==Definition==
−
Mathematicians of the same University were not so arrogant and used the symbol of [[integral]] and the [[Moebius surface]] at their logo, see second picture of the figure at left.
  
   
−
That time, researchers did not have computational facilities for evaluation of such functions, but
  
−
the <math>\sqrt{\exp}</math> was more lucky than the <math>~\sqrt{!~}~~</math>; at least the existence of [[holomorphic function]]
 
−
<math>\varphi</math> such that <math>\varphi(\varphi(z))=\exp(z)</math> has been declared <ref name="kneser"/>. Actually, for his proof, Kneser had constructed the [[superfunction]] of exp and corresponding [[Abel function]] <math>\mathcal{X}</math>, satisfying the [[Abel equation]]
  
−
: <math>\mathcal{X}(\exp(z))=\mathcal{X}(z)+1</math> .
  
−
The [[inverse function]], id est <math>F=\mathcal \chi^{-1}</math> is an [[entire function|entire]] super-exponential, although it is not real at the real axis; it cannot be interpreted as [[tetration]], because the condition <math>F(0)=1</math> cannot be realized for the entire super-exponential. The [[real function|real]] <math>\sqrt{\exp}</math> can be constructed with the [[tetration]] (which is also a superexponential), and the real <math>\sqrt{\rm Factorial}</math> can be constructed with the [[SuperFactorial]]. The plots of <math>\sqrt{\rm Factorial}~</math> and <math>\sqrt{\exp}~</math> in the compex plane are shown in the right hand side figure.
  
   
−
==Extensions==
  
−
The recurrent formula of the preamble can be written as equations
  
   
  +
The integer values of the argument, the superfunction has simple sense.
−
\(F(z\!+\!1)=T(F(z)) ~ \forall z\in \mathbb{N} : z>0\)
  
   
  +
For some constant \(t\), the superfunction \(F\) can be expressed as follows: &nbsp;
−
\(F(1)=t\)
  
  +
\(\displaystyle {{F(z)} \atop \,} {= \atop \,}
  +
{T^z(t) \atop \,} {= \atop \,}
  +
{{\underbrace{T\Big(T\big(... T(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}T\!
  +
\!\!\!\!\!}}\)
   
  +
In general case, the [[Superfunction]] can be defined as follows:
−
Instead of the last equation, one could write
  
   
−
\(F(0)=t\)
  
−  
−
and extend the range of definition of superfunction <math>F</math> to the non-negative integers. Then, one may postulate
 
−  
−
\(F(-1)=T^{-1}(t)\)
  
−  
−
and extend the range of validity to the negative integer values, at least while the inverse of the Transfer function is holomorphic.
  
−
For example,
  
−  
−
\(F(-2)=T^{-2}(t)=T^{-1}\Big(T^{-1}(t)\Big)\)
  
−  
−
and so on. However, the inverse function may happen to be not defined for some values of <math>t</math>.
  
−  
−
The [[tetration]] is considered as super-function of exponential for some real base <math>b</math>; in this case,
  
−  
−
\(T=\exp_{b}\)
  
−  
−
then, at \(t=1\),
  
−  
−
\(F(-1)=\log_b(1)=0 ~ ~, ~ ~ \mathrm{but}\)
 
−  
−
\(F(-2)=\log_b(0)~ \mathrm{~is~ not~ defined}\)
  
−  
−
For extension to non-integer values of the argument, superfunction should be defined in different way.
  
−  
−
==Definition==
  
 
For complex numbers <math>~p~</math> and <math>~q~</math>, such that <math>~p~</math> belongs to some connected domain <math>D\subseteq \mathbb{C}</math>,<br>
  
For complex numbers <math>~p~</math> and <math>~q~</math>, such that <math>~p~</math> belongs to some connected domain <math>D\subseteq \mathbb{C}</math>,<br>
 
<!-- <math>a \!\mapsto\! b</math> !-->
  
<!-- <math>a \!\mapsto\! b</math> !-->
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function <math> F </math>, [[holomorphic]] on domain <math>D</math>, such that
  
function <math> F </math>, [[holomorphic]] on domain <math>D</math>, such that
   
−
\(F(z\!+\!1)=T(F(z)) ~ \forall z\in D : z\!+\!1 \in D\)
 +
\[F(z\!+\!1)=T(F(z)) ~ \forall z\in D : z\!+\!1 \in D\]
   
−
\(F(p)=q\)
 +
\[F(p)=q\]
   
 
==Uniqueness==
  
==Uniqueness==
−
In general, the super-function is not unique.
  
−
For a given base function <math>T</math>, from given <math>(p\mapsto q)</math> superfunction <math>F</math>, another <math>(p \mapsto q)</math> superfunction <math>\tilde F</math> could be constructed as
  
−
: <math>\tilde F(z)=F(z+\mu(z))</math>
  
−
where <math>\mu</math> is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that <math> \mu(p)=0 </math>.
  
   
  +
In general, for a given holomorphic [[transfer function]] \(T\),
−
The modified superfunction may have narrowed range of holomorphism.
  
  +
the [[superfunction]] \(F\) is not unique, even is some specific value \(F(0)\) is fixed.
−
The narrower is the range of holomorphism, the wider is variety of superfunctions allowed <ref name="walker">
  
−
http://www.jstor.org/stable/2938713
  
−
P.Walker
  
−
Infinitely differentiable generalized logarithmic and exponential functions
  
−
[[Mathematics of computation]], '''196''' (1991), 723-733
  
−
</ref>.
  
   
  +
Other superfunctions \(f\) can be constructed as follows:
−
At large enough range of holomorphism, the super-function is expected to be unique, for each specific
  
−
specific [[transfer function]] <math>T</math>. In particular, the <math>(C, 0\mapsto 1)</math> super-function of
  
−
<math>\exp_b</math>, for <math>b>1</math>, is called [[tetration]] and is believed to be unique at least for
 
−
<math>C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \}</math>; for the case <math> b>\exp(1/\mathrm{e})</math>, see <ref name="kouznetsov">
  
−
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
  
−
(preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf )
  
−
D.Kouznetsov.
  
−
Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.
 
−
[[Mathematics of Computation]],
  
−
'''78''' (2009) 1647-1670
  
−
<!--10.1090/S0025-5718-09-02188-7!-->
  
−
</ref>.
  
   
  +
\[
−
==Examples==
  
  +
f(z)=F(z+\theta(z))
−
The short version of the [[table of superfunctions]] is suggested in
 
  +
\]
−
<ref name="superfactorial">
  
  +
where \(\theta\) is any periodic holomorphic periodic function with period unity.
−
http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf
  
−
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. [[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12.
 
−
</ref>. A little bit more extended table is avilable at [[TORI]]
  
−
<ref name="toritable">
  
−
http://tori.ils.uec.ac.jp/TORI/index.php/Table_of_superfunctions
  
−
</ref>.
  
−
Some superfunctions can be expressed with elementary functions,
 
−
they are used without to mention that they are superfunctions.
  
−
For example, for the transfer function "++", which means unity increment,
  
−
the superfunction is just addition of a constant.
  
   
  +
The \(\theta\) can be represented as sum of sinusoids,
−
===Addition===
  
−
Chose a [[complex number]] <math>c</math> and define function
 
−
<math>\mathrm{add}_c</math> with relation
 
−
<math>\mathrm{add}_c(z)=c\!+\!z ~ \forall z \in \mathbb{C}</math>
  
−
<!-- , where <math>c\in \mathbb{C}</math> is constant.!-->.
  
−
Define function
 
−
<math>\mathrm{mul_c}</math> with relation
 
−
<math>\mathrm{mul_c}(z)=c\!\cdot\! z ~ \forall z \in \mathbb{C}</math>.
  
   
  +
\[
−
Then, function <math>~\mathrm{mul_c}~</math> is '''superfunction''' (<math>~0</math> to <math>~ c~</math>)
  
  +
\theta(z)= \sum_{k=1}^{\infty} a_k \sin(2\pi k z)
−
of function <math>~\mathrm{add_c}~</math> on <math>~\mathbb{C}~</math>.
  
  +
\]
   
  +
with some real coefficients \(a\); such a representation converts the
−
===Multiplication===
  
  +
real-holomorphic superfunction \(F\) to another
−
Exponentiation <math>\exp_c</math> is superfunction (from 1 to <math>c</math>) of function <math>\mathrm{mul}_c </math>.
  
  +
real-holomorphic superfunction \(f\),
  +
and even preserves the value at zero;
  +
\[
  +
f(0)=F(0)
  +
\]
   
  +
However, each sin in this representation shows fast growth in the direction of imaginary axis.
−
===Quadratic polynomials===
  
−
Let the transfer function \(T\) be defined with \(T(z)=2 z^2-1\).
  
−
Then,
 
−
\(F(z)=\cos( \pi \cdot 2^z) \) is a
 
−
\((\mathbb{C},~ 0\! \rightarrow\! 1)\) superfunction of \(T\).
  
   
  +
If we fix the asymptotic behavior at \(\pm \mathrm i\infty\)
−
Indeed,
 
   
  +
then we may hope the solution \(F\) of the transfer equation to be unique.
−
\(F(z\!+\!1)=\cos(2 \pi \cdot 2^z)=2\cos(\pi \cdot 2^z)^2 -1 =T(F(z))\)
  
   
  +
In particular for the [[natural Tetration]],
−
and
  
  +
it is postulated, that at \(\pm \mathrm i\infty\), this the superfunction approaches
  +
the [[fixed point]]s \(L\) and \(L^*\) of natural logarithm;
  +
\[
  +
L = -\mathrm{ProductLog}(-1)^* \approx 0.3181315052047641+ 1.3372357014306895 \mathrm{I}
  +
\]
  +
This requirement not only provides the uniqueness of the [[tetration]] <ref>
  +
https://link.springer.com/article/10.1007/s00010-010-0021-6 <br>
  +
http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf
  +
H.Trappmann, D.Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair Aequationes Mathematicae, v.81, p.65-76 (2011)
  +
</ref>,
  +
but also gives the way for the efficient evaluation through the [[Cauchi intergal]]
  +
and/or the asymptotic expansion of [[tetration]] at \(\pm \mathrm i\infty\)
  +
<ref>
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  +
</ref> and the fast precise numerical implementation
  +
<ref>
  +
http://www.vmj.ru/articles/2010_2_4.pdf <br>
  +
http://mizugadro.mydns.jp/PAPERS/2009vladie.pdf
  +
D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  +
</ref>.
   
  +
==Other meanings of the words==
−
\(F(0)=\cos(2\pi)=1\)
  
  +
<div class="thumb tright" style="float:right;margin:-30px 12px 0px 8px">
  +
{{picx|Superfunction_icon.jpg|80px}} <small><center>other "[[superfunction]]s" <ref>
  +
https://endless-sport.co.jp/products/suspension/index_superfunction.html Functionメーカー別適合表 ホーム> サスペンショントップ Super Function / スーパーファンクション（サーキット向け）(2020).
  +
</ref><ref>
  +
https://www.pythonforbeginners.com/super/working-python-super-function
  +
Working with the Python Super Function.
  +
Last Updated: May 20, 2020.
  +
Python 2.2 saw the introduction of a built-in function called “super,” which returns a proxy object to delegate method calls to a class – which can be either parent or sibling in nature. <!--
  +
That description may not make sense unless you have experience working with Python, so we’ll break it down.
  +
Essentially, the super function can be used to gain access to inherited methods – from a parent or sibling class – that has been overwritten in a class object.!-->
  +
</ref></small></center>
  +
</div>
  +
<div class="thumb tright" style="float:right;margin:2px 12px 0px 8px">
  +
{{picx|Nestbird1.png|88px}}{{picx|107253_original.jpg|88px}}<small><center> [[Nest]] ; [[Superpower]]</small></center></div>
   
  +
Mathematics often borrow terms from the common life. This may cause confieions.
−
In this case, the superfunction \(F\) is periodic; its period
  
  +
In particular this applies to terms [[Superfunciton]] and [[Nest]]. Few examples are shown in figures at right.
   
  +
Not all supperfunctions are holomorphic.
−
\(\tau=\frac{2\pi}{\ln(2)} \mathrm{i}\approx 9.0647202836543876194 \!~i \)
  
   
  +
Some [[Nest]]s may be difficult to implement in [[Mathematica]].
−
and the superfunction approaches unity also in the negative direction of the real axis,
  
   
  +
Even term [[Superpower]] may have different meanings.
−
\(\lim_{x\rightarrow -\infty} F(x)=1\)
  
   
  +
In [[TORI]] there terms are not used in sense shown in figures at right.
−
The example above and the two examples below are suggested at
  
−
<ref name="mueller">Mueller. Problems in Mathematics. http://www.math.tu-berlin.de/~mueller/projects.html </ref>
  
   
  +
==References==
−
In general, the transfer function \(T\) has no need to be [[entire function]].
  
  +
{{ref}}
−
Here is the example with [[meromorphic function]] \(T\).
 
−
Let
  
   
  +
http://www.proofwiki.org/wiki/Definition:Superfunction
−
\(T(z)=\frac{2z}{1-z^2} ~ \forall z\in D~\); \(~ D=\mathbb{C} \backslash \{-1,1\}\)
  
   
  +
http://en.citizendium.org/wiki/Superfunction
−
Then, function
  
   
  +
2014. https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
−
\(F(z)=\tan(\pi 2^z)\)
  
  +
Dmitrii Kouznetsov. Суперфункции. Lambert Academic Publishing, 2014. (In Russian)
   
  +
2016.
−
is \((C, 0\! \mapsto\! 0)\) superfunction of function \(T\), where
  
  +
https://doi.org/10.11568/kjm.2016.24.1.81
−
\(C\) is the set of complex numbers except singularities of function \(F\).
  
  +
W.Paulsen, “FINDING THE NATURAL SOLUTION TO f(f(x)) = exp(x),” Korean Journal of Mathematics, vol. 24, no. 1, pp. 81–106, Mar. 2016.
−
For the proof, the trigonometric formula
  
−
\(\displaystyle
  
−
\tan(2 \alpha)=\frac{2 \tan(\alpha)}{1-\tan(\alpha)^2}~~
  
−
\forall \alpha \in \mathbb{C} \backslash \{\alpha\in \mathbb{C} : \cos(\alpha)=0 || \sin(\alpha)=\pm \cos(\alpha) \}
  
−
\)
  
−
can be used at <math>\alpha=\pi 2^z </math>, that gives
  
   
  +
2017.
−
\(\displaystyle
  
  +
https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1
−
T(F(z))=\frac{2 \tan(\pi 2^z)}{1-\tan(\pi 2^z)} = \tan(2 \pi 2^z)=F(z+1)\)
  
  +
S.Cowgill. Exploring Tetration in the Complex Plane.
−
<!-- However, such function <math>F(z)</math> allows the holomrphic extension to values, where <math>cos(\pi 2^z)=0</math>,
  
  +
Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680.
−
setting it to zero in these points, but it has singularities, poles, at <math>2^z=\frac{1+2n}{4}</math> for integer <math>n</math>.
  
−
!-->
  
   
  +
2017.
−
===Algebraic function===
  
  +
https://doi.org/10.1007/s10444-017-9524-1
−
In the similar way one can consider the transfer function
 
  +
W.Paulsen, S.Cowgill, Solving F(z + 1) = b ^ F(z) in the complex plane. Adv Comput Math 43, 1261–1282 (2017).
   
  +
2018.
−
\(T(z)=2z \sqrt{1-z^2}\)
  
  +
https://doi.org/10.1080/10236198.2017.1307350
  +
M.H. Hooshmand. (2018) Ultra power of higher orders and ultra exponential functional sequences. Journal of Difference Equations and Applications.
  +
Volume 24, 2018 - Issue 5: Special Issue: European Conference on Iteration Theory 2016. Special Issue Editors: Marek Cezary Zdun & Jaroslav Smital.
   
  +
2019.
−
and
  
  +
https://doi.org/10.1007/s10444-018-9615-7
  +
W.Paulsen. Tetration for complex bases. Adv Comput Math 45, 243–267 (2019).
   
  +
2020.
−
\(F(z)=\sin(\pi 2^z)\)
  
  +
https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3
  +
D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
   
−
which is \((C,~ 0\!\rightarrow \!0)\) superfunction of \(H\) for
  
−
\(C= \{z\in \mathbb C : \Re( \cos(\pi 2^z))>0 \}\).
  
−  
−
===Exponentiation===
  
−
Let
 
−
<math>b>1</math>,
 
−
<math>T(z)= \exp_b(z)</math>,
  
−
<math> C= \{ z \in \mathbb{C} : \Re(z)>-2 \}</math>.
 
−
Then, [[tetration]] <math> \mathrm{tet}_b </math>
  
−
is a <math>(C,~ 0\! \rightarrow\! 1)</math> superfunction of <math>\exp_b</math>.
  
−  
−
==Abel function==
  
−
Inverse of superfunction \(G=F^{-1}\) is called the [[Abel function]]; within some domain, it satisfies the Abel equation
  
−  
−
\(T(G(z))=G(z)+1\)
  
−  
−
==Applications of superfunctions and the Abel functions==
  
−  
−
Superfunctions, usially the [[tetration|superexponential]]s, are proposed as a fast-growing function for an
  
−
upgrade of the [[floating point]] representation of numbers in computers. Such an upgrade would greatly extend the
 
−
range of huge numbers which are still distinguishable from infinity.
  
−  
−
Other applications refer to the calculation of fractional iterates
 
−
(or fractional power) of a function. Any holomorphic function can be declared as a "transfer function", then its superfunctions and
 
−
corresponding Abel functions can be considered.
  
−
===Nest===
  
−
The \(c\)th iteration of some function \(f\) can be expressed through the [[superfunction]] \(F\) and the [[Abel function]] \(G=F^{-1}\):
  
−
:\( T^c(z)=F(c+G(z))\)
  
−
In [[Mathematica]], there already exist the special name for the operation, that could evaluate such a expression. It is called [[Nest]].
  
−
<ref name="nest">
  
−
http://reference.wolfram.com/mathematica/ref/Nest.html
  
−
</ref>. This function has 3 arguments.
  
−
The first argument indicates the name of the function.
 
−
The second argument indicates the initial value.
  
−
The third (and last) argument indicates the number of iterations.
  
−
Then, the iteration of function \(T\) can be written as follows:
  
−
:\( T^c(z)=\mathrm {Nest}[T,z,c]\)
  
−  
−
Unfortunately, in the version "Mathematica 8", the implementation of the [[Nest]] has serious restrictions: the number of iterations should allow the simplification to an integer constant.
  
−
The intents to call function [[Nest]] with any other expression as the last argument cause the error messages. One may hope, in the future versions of Mathematica this bug will be corrected.
  
−  
−
The [[Table of superfunctions|table of known superfunctions]] and the corresponding [[Abel function]]s
 
−
(similar to that suggested in <ref name="superfactorial">
  
−
http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf
  
−
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. [[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12.
 
−
</ref>)
  
−
could be loaded in Mathematica in a manned similar to that the [[table of integrals]] is loaded. This would allow the correct implementation of [[Nest]] for the case of non–integer number of iterations.
  
−  
−
===Transition from a function to its inverse function===
  
−
[[Image:Expc.jpg|200px|left|thumb|<math>\exp^c(x)</math> versus <math>x</math> for various <math>c</math>]]
  
−
[[Image:Sqrt(exp)(z).jpg|600px|right|thumb|<math>\exp^c</math> in the complex plane for various <math>c</math>]]
  
−  
−
A [[superfunction]] <math>S</math> allows to calculate the fractional [[iteration]] <math>H^c</math> of some transfer function <math>H</math>. Once the superfunction <math>S</math> and the [[Abel function]] <math>A=S^{-1}</math> are established,
 
−
the fractional iteration can be defined as
 
−
<math>H^c(z)=S(c+A(z))</math>. Then, as <math>c</math> changes from 1 to <math>-1</math>, the holomorphic transition from function <math>H</math> to <math>H^{-1}</math> is relalised. The figure at left shows an example of transition from
 
−
<math>\exp^{1}\!=\!\exp </math> to
 
−
<math>\exp^{\!-1}\!=\!\ln </math>.
  
−
Function <math>\exp^c</math> versus real argument is plotted for
  
−
<math>c=2,1,0.9, 0.5, 0.1, -0.1,-0.5, -0.9, -1,-2</math>. The [[tetration]]al and ArcTetrational were used as superfunction
 
−
<math>F</math>
  
−
and Abel function <math>G</math> of the exponential.
  
−
The figure at right shows these functions in the [[complex plane]].
  
−
At non-negative integer number of [[iteration]], the iterated exponential is [[entire function]]; at non-integer values, it has two [[branch points]], thich correspond to the [[fixed points]] <math>L</math> and
  
−
<math>L^*</math> of natural logarithm. At <math>c\!\ge\! 0</math>, function <math>\exp^c(z)</math> remains [[holomorphic function|holomorphic]] at least in the strip <math>|\Im(z)|<\Im(L)\approx 1.3 </math> along the real axis.
  
−  
−
===Nonlinear Optics===
  
−
In the investigation of the nonlinear response of optical materials,
  
−
the sample is supposed to be optically thin, in such a way,
  
−
that the intensity of the light does not change much as it goes through.
  
−
Then one can consider, for example, the absorption as function of the intensity.
  
−
However, at small variation of the intensity in the sample,
  
−
the precision of measurement of the absorption as function of intensity is not good.
  
−
The reconstruction of the superfunction from the Transfer Function allows to work with
  
−
relatively thick samples, improving the precision of measurements. In particular, the
  
−
Transfer Function of the similar sample, which is half thiner,
  
−
could be interpreted as the square root (id est, half-iteration) of the Transfer Function of the initial sample.
  
−  
−
Similar example is suggested for a nonlinear optical fiber
 
−
<ref name="kouznetsov">
  
−
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
  
−
(preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf )
  
−
D.Kouznetsov.
  
−
Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.
 
−
[[Mathematics of Computation]],
  
−
'''78''' (2009) 1647-1670
  
−
<!--10.1090/S0025-5718-09-02188-7!-->
  
−
</ref>.
  
−
In particular, the [[Tania function]] can be evaluated using the [[Doya function]] and the [[regular iteration]]
 
−
<ref name="2013or">
  
−
http://link.springer.com/article/10.1007/s10043-013-0058-6
  
−
Dmitrii Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.
  
−
Preprint: http://mizugadro.mydns.jp/PAPERS/2013orSuper.pdf
  
−
</ref><!--<ref name="sinapo"> http://mizugadro.mydns.jp/PAPERS/2011singapo.pdf
  
−
D.Kouznetsov. Transfer function of an amplifier and characterization of Materials. Singapore, 2011. (slideshow)</ref>!-->
  
−
at the fixed point \(0\), as \(\mathrm{Doya}_t(0)\!=\!0\). The same method can be applied also to other [[transfer funciton]]s, even if they cannot be easy represented through the [[special function]]s. This may refer, for example, to the experimentally–measured transfer function, that has no need to coincide with the [[Doya function]].
  
−  
−
===Nonlinear Acoustics===
  
−
It may have sense to characterize the nonlinearities in the
  
−
attenuation of shock waves in a homogeneous tube. This could find an application in some
  
−
advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves
 
−
without to disturb the flux of the gas. Again, the analysis of the nonlinear response,
 
−
id est, the Transfer Function,
  
−
may be boosted with the superfunction.
  
−  
−
===Vaporization and condensation===
  
−
<!--For the separation of isotopes due to the different pressure of the saturated vapor for different components,!-->
  
−
In analysis of condensation, the growth (or vaporization) of a small drop of liquid can be considered,
  
−
as it diffuses down through a tube with some uniform concentration of vapor.
  
−
In the first approximation, at fixed concentration of the vapor,
  
−
the mass of the drop at the output end can be interpreted as the
  
−
Transfer Function of the input mass.
  
−
The square root of this Transfer Function will characterize the tube of half length.
  
−  
−
===Snow avalanche===
  
−
The mass of a snowball, that rolls down from the hill,
  
−
can be considered as a function of the path it already have passed. At fixed length of this path
  
−
(that can be determined by the altitude of the hill) this mass can be considered also as a Transfer Function of the input mass. The mass of the snowball could be measured at the top of the hill and at thе bottom, giving the Transfer Function; then, the mass of the snowball as a function of the length it passed is superfunction.
  
−  
−
===Operational element===
  
−
If one needs to build-up an operational element with some given transfer function <math>H</math>,
  
−
and wants to realize it as a sequential connection of a couple of identical operational elements, then, each of these two elements should have transfer function
 
−
<math> h=\sqrt{T}</math>. Such a function can be evaluated through the superfunction and the Abel function of the transfer function <math>T</math>.
  
−  
−
The operational element may have any origin: it can be realized as an electronic microchip,
  
−
or a mechanical couple of curvilinear grains), or some asymmetric U-tube filled with different liquids, and so on.
  
−  
−
==References==
  
−
<references/>
  
−
\
  
−
http://www.proofwiki.org/wiki/Definition:Superfunction
  
−  
−
http://en.citizendium.org/wiki/Superfunction
  
   
 
==Keywords==
  
==Keywords==
−
[[Abel function]],
 +
«[[Abel function]]»,
−
[[Iteration]],
 +
«[[Iteration]]»,
−
[[Regular iteration]],
 +
«[[Regular iteration]]»,
−
[[SuperFactorial]],
 +
«[[SuperFactorial]]»,
−
[[Superfunctions]],
 +
«[[Superfunctions]]»,
−
[[Transfer function]],
 +
«[[Transfer function]]»,
−
[[Transfer equation]],
 +
«[[Transfer equation]]»,
−
[[Tetration]],
 +
«[[Tetration]]»,
−
[[Tania function]],
 +
«[[Tania function]]»,
−
[[Tania function]],
 +
«[[Tania function]]»,
−
[[Shoka function]],
 +
«[[Shoka function]]»,
−
[[SuZex]],
 +
«[[SuZex]]»,
−
[[SuTra]]
 +
«[[SuTra]]».
   
−
[[Суперфункции]]
 +
«[[Суперфункции]]»,
   
 
[[Category:English]]
  
[[Category:English]]
Line 395: Line 265:
 
[[Category:Mathematics]]
  
[[Category:Mathematics]]
 
[[Category:Superfunction]]
  
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]

Latest revision as of 15:10, 18 August 2025


For some holomorphic function \(T\) (called «Transfer function»), the Superfunction \(F\) is solution of the transfer equation

\[ F(z\!+\!1) = T(F(z)) \]

Formalism of superfunctions is described in special book «Superfunctions» (2020) [1][2].
There is also Russian version «Суперфункции» (2014) [3].

Superfunctions allow to constrict non-integer iterates of holomorphic functions; in particular, the square root of factorial \(\ \phi\!=\!\sqrt{\ !\ }\ \) as solution [4] of equation

QFactorialQexp.jpg
Complex maps \(\ p\!+\!\mathrm iq=\phi(z)\ \) and \(\ p\!+\!\mathrm iq\!=\!\varphi(z)\ \) [4]

\[ \phi(\phi(z))=z! \] and the «square root of exponential» [5] \(\ \varphi\!=\!\sqrt{\exp}\ \) as solution of equation \[ \varphi(\varphi(z))=\exp(z) \]

Maps of functions \(\phi\) and \(\varphi\) are shown in first figure.

  Sqrt(factorial)LOGOintegralLOGO.jpg

  HellmuthKneserPhotoNormal.jpeg

LOGO \(\sqrt{\ !\ }\) and H.Kneser (\(\sqrt{\exp}\)) [4][6]

For a given transfer function \(T\) and its superfunction \(F\), the inverses function \(G\!=\!F^{-1}\) is the Abelfunction. While for the Transfer function \(T\), the Superfunction \(F\) and the Abelfunction \(G\) are established, the \(n\)th iterate of the Transfer function can be expressed as follows:

\[ T^n(z)=F(n+G(z)) \]

In this expression, number \(n\) of iterate has no need to be integer; it can be real or even complex.

In such a way, with the superfunction and the Abelfunction, the transfer function can be iterated arbitrary times. To century 21, this property seems to provide the main application of superfunctions.

History

Functions \(\sqrt{\ !\ }\) and \(\sqrt{\exp} \) are mentioned century 20 as LOGO pf Physics Department of the Moscow State University [7][8][9] and as holomorphic solution of \(\varphi(\varphi(z))=\exp(z)\) [4] suggested by Hellmith Kneser, see pictures above.

In century 20, before construction of formalism of superfunctions, these functions were not implemented; the complex maps of these functions were not plotted, although many publications about non-integer iterates of functions are observed. The first robust implementations of non-trivial superfunctions and ablefunctions appear in years 2009-2010 [4]

Definition

The integer values of the argument, the superfunction has simple sense.

For some constant \(t\), the superfunction \(F\) can be expressed as follows:   \(\displaystyle {{F(z)} \atop \,} {= \atop \,} {T^z(t) \atop \,} {= \atop \,} {{\underbrace{T\Big(T\big(... T(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}T\! \!\!\!\!\!}}\)

In general case, the Superfunction can be defined as follows:

For complex numbers \(~p~\) and \(~q~\), such that \(~p~\) belongs to some connected domain \(D\subseteq \mathbb{C}\),
superfunction (from \(p \mapsto q\)) of holomorphic function \(~T~\) on domain \(C \in \mathbb C\) is function \( F \), holomorphic on domain \(D\), such that

\[F(z\!+\!1)=T(F(z)) ~ \forall z\in D : z\!+\!1 \in D\]

\[F(p)=q\]

Uniqueness

In general, for a given holomorphic transfer function \(T\), the superfunction \(F\) is not unique, even is some specific value \(F(0)\) is fixed.

Other superfunctions \(f\) can be constructed as follows:

\[ f(z)=F(z+\theta(z)) \] where \(\theta\) is any periodic holomorphic periodic function with period unity.

The \(\theta\) can be represented as sum of sinusoids,

\[ \theta(z)= \sum_{k=1}^{\infty} a_k \sin(2\pi k z) \]

with some real coefficients \(a\); such a representation converts the real-holomorphic superfunction \(F\) to another real-holomorphic superfunction \(f\), and even preserves the value at zero; \[ f(0)=F(0) \]

However, each sin in this representation shows fast growth in the direction of imaginary axis.

If we fix the asymptotic behavior at \(\pm \mathrm i\infty\)

then we may hope the solution \(F\) of the transfer equation to be unique.

In particular for the natural Tetration, it is postulated, that at \(\pm \mathrm i\infty\), this the superfunction approaches the fixed points \(L\) and \(L^*\) of natural logarithm; \[ L = -\mathrm{ProductLog}(-1)^* \approx 0.3181315052047641+ 1.3372357014306895 \mathrm{I} \] This requirement not only provides the uniqueness of the tetration [10], but also gives the way for the efficient evaluation through the Cauchi intergal and/or the asymptotic expansion of tetration at \(\pm \mathrm i\infty\) [11] and the fast precise numerical implementation [12].

Other meanings of the words

Superfunction icon.jpg
other "superfunctions" [13][14]
Nestbird1.png107253 original.jpg
Nest ; Superpower

Mathematics often borrow terms from the common life. This may cause confieions. In particular this applies to terms Superfunciton and Nest. Few examples are shown in figures at right.

Not all supperfunctions are holomorphic.

Some Nests may be difficult to implement in Mathematica.

Even term Superpower may have different meanings.

In TORI there terms are not used in sense shown in figures at right.

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics – 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
  3. ↑ https://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2014
  4. ↑ 4.0 4.1 4.2 4.3 4.4 https://link.springer.com/article/10.3103/S0027134910010029
    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. (Russian version: p.8-14)
  5. ↑ http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002175851&physid=phys63#navi Hellmuth Kneser. Reelle analytische Lösungen der Gleichung \( \varphi(\varphi(x))=\mathrm e^x \) und verwandter Funktionalgleichungen. Journal für die reine und angewandte Mathematik / Zeitschriftenband (1950) / Artikel / 56 - 67
  6. ↑ https://opc.mfo.de/detail?photo_id=7607 On the Photo: Kneser, Hellmuth Location: Oberwolfach Author: Danzer, Ludwig (photos provided by Danzer, Ludwig) Source: L. Danzer, Dortmund Year: 1958 Copyright: L. Danzer, Dortmund Photo ID: 7607
  7. ↑ (not available: http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf ) V.P.Kandidov. About the time and myself. (In Russian) 2007.05.18. .. 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. ..
  8. ↑ https://www.nkj.ru/archive/articles/1023/ V.Sadovnichi. 250 anniversary of the Moscow State University. (In Russian) В. Садовничий. ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250! НАУКА И ЖИЗНЬ, №1, 2005. .. На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее. ..
  9. ↑ Logo of the Physics Department of the Moscow State University. (In Russian); http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml
  10. ↑ https://link.springer.com/article/10.1007/s00010-010-0021-6
    http://mizugadro.mydns.jp/PAPERS/2011uniabel.pdf H.Trappmann, D.Kouznetsov. Uniqueness of holomorphic Abel functions at a complex fixed point pair Aequationes Mathematicae, v.81, p.65-76 (2011)
  11. ↑ http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  12. ↑ http://www.vmj.ru/articles/2010_2_4.pdf
    http://mizugadro.mydns.jp/PAPERS/2009vladie.pdf D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  13. ↑ https://endless-sport.co.jp/products/suspension/index_superfunction.html Functionメーカー別適合表 ホーム> サスペンショントップ Super Function / スーパーファンクション（サーキット向け）(2020).
  14. ↑ https://www.pythonforbeginners.com/super/working-python-super-function Working with the Python Super Function. Last Updated: May 20, 2020. Python 2.2 saw the introduction of a built-in function called “super,” which returns a proxy object to delegate method calls to a class – which can be either parent or sibling in nature.

http://www.proofwiki.org/wiki/Definition:Superfunction

http://en.citizendium.org/wiki/Superfunction

2014. https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 Dmitrii Kouznetsov. Суперфункции. Lambert Academic Publishing, 2014. (In Russian)

2016. https://doi.org/10.11568/kjm.2016.24.1.81 W.Paulsen, “FINDING THE NATURAL SOLUTION TO f(f(x)) = exp(x),” Korean Journal of Mathematics, vol. 24, no. 1, pp. 81–106, Mar. 2016.

2017. https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1 S.Cowgill. Exploring Tetration in the Complex Plane. Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680.

2017. https://doi.org/10.1007/s10444-017-9524-1 W.Paulsen, S.Cowgill, Solving F(z + 1) = b ^ F(z) in the complex plane. Adv Comput Math 43, 1261–1282 (2017).

2018. https://doi.org/10.1080/10236198.2017.1307350 M.H. Hooshmand. (2018) Ultra power of higher orders and ultra exponential functional sequences. Journal of Difference Equations and Applications. Volume 24, 2018 - Issue 5: Special Issue: European Conference on Iteration Theory 2016. Special Issue Editors: Marek Cezary Zdun & Jaroslav Smital.

2019. https://doi.org/10.1007/s10444-018-9615-7 W.Paulsen. Tetration for complex bases. Adv Comput Math 45, 243–267 (2019).

2020. https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3 D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.


Keywords

«Abel function», «Iteration», «Regular iteration», «SuperFactorial», «Superfunctions», «Transfer function», «Transfer equation», «Tetration», «Tania function», «Tania function», «Shoka function», «SuZex», «SuTra».

«Суперфункции»,

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