Difference between revisions of "Superfunction"

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#redirect[[Superfunction]]
  
  +
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 [[superfunction]]s is described in special book «[[Superfunctions]]» (2020)
  +
<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
  +
</ref>
  +
</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:
  +
  +
\[
  +
T^n(z)=F(n+G(z))
  +
\]
  +
  +
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 [[superfunction]]s.
  +
==History==
  +
  +
Functions \(\sqrt{\ !\ }\) and \(\sqrt{\exp} \) are mentioned century 20
  +
as LOGO pf Physics Department of the Moscow State University
  +
<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.
  +
.. 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. ..
  +
</ref><ref name="naukai">
  +
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. ..</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
  +
</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 first robust implementations of non-trivial superfunctions and ablefunctions
  +
appear in years 2009-2010
  +
<ref name="f"/>
  +
  +
==Definition==
  +
  +
  +
  +
The integer values of the argument, the superfunction has simple sense.
  +
  +
For some constant \(t\), the superfunction \(F\) can be expressed as follows: &nbsp;
  +
\(\displaystyle {{F(z)} \atop \,} {= \atop \,}
  +
{T^z(t) \atop \,} {= \atop \,}
  +
{{\underbrace{T\Big(T\big(... T(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}T\!
  +
\!\!\!\!\!}}\)
  +
  +
In general case, the [[Superfunction]] can be defined as follows:
  +
  +
For complex numbers <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> !-->
  +
superfunction (from <math>p \mapsto q</math>) of [[holomorphic function]] <math>~T~</math> on domain <math>C \in \mathbb C</math> is
  +
function <math> F </math>, [[holomorphic]] on domain <math>D</math>, such that
  +
  +
\[F(z\!+\!1)=T(F(z)) ~ \forall z\in D : z\!+\!1 \in D\]
  +
  +
\[F(p)=q\]
  +
  +
==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 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==
  +
<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 particular this applies to terms [[Superfunciton]] and [[Nest]]. Few examples are shown in figures at right.
  +
  +
Not all supperfunctions are holomorphic.
  +
  +
Some [[Nest]]s 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==
  +
{{ref}}
  +
  +
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]]»,
  +
  +
[[Category:English]]
  +
[[Category:Iterate]]
  +
[[Category:Iteration]]
  +
[[Category:Mathematics]]
  +
[[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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