Difference between revisions of "Superfunctions"
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+  [[Superfunctions]] is book about evaluation of [[abelfunction]]s, [[superfunction]]s, and the noninteger [[iterate]]s of holomorphic functions. 

−  [[File:437front.jpg300pxthumbFront cover of the Book]] 

+  In particular, results for [[tetration]], [[arctetration]] and [[iterate]]s of [[exponential]] are presented, ISBN 6202672862 

−  [[File:437back.jpg300pxthumbback cover of the Book]] 

+  <ref> 

−  [[File:Tetreal10bx10d.png300pxthumbExample of the explicit plot from the Book: plot of [[tetration]], 

+  https://www.morebooks.de/store/gb/book/superfunctions/isbn/9786202672863 

−  $y=\mathrm{tet}_b(x)$ versus $x$ for various $b\!>\!1$, Figure 17.1]] 

+  Superfunctions: Noninteger iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics Paperback – July 28, 2020 

−  [[File:Ausintay40t50.jpg300pxthumbExample of the [[complex map]] from the Book: map of the truncated Taylor expansion of abelsine [[AuSin]], Figure 22.2 ]] 

+  by Dmitrii Kouznetsov (Author)// 

−  [[File:Oglavleni.png150pxthumbLiterature cited in the Book, Figure 21.1]] 

+  Publisher : LAP LAMBERT Academic Publishing (July 28, 2020)// 

−  [[File:Bookwormbw.png150pxthumbLiterature not cited in the Book, Figure 21.3]] 

+  Language: : English// 

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

+  Paperback : 328 pages// 

+  ISBN10 : 6202672862// 

+  ISBN13 : 9786202672863// 

+  Item Weight : 1.18 pounds// 

+  Dimensions : 5.91 x 0.74 x 8.66 inches 

+  </ref><ref> 

+  https://www.morebooks.de/store/gb/book/superfunctions/isbn/9786202672863 <br> 

+  Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020. 

+  </ref> 

+  https://www.morebooks.de/store/gb/book/superfunctions/isbn/9786202672863<br> 

−  The English version is loaded as http://mizugadro.mydns.jp/BOOK/444.pdf 

+  Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020. 

+  <div class="thumb tright" style="float:right;width:384px"> 

+  <div style="width:400px"> 

+  [[File:9786202672863full.jpg400px]] 

+  Cover at [[Lambert Academic Publishing]], 2020 

+  </div></div> 

+  The Book is loaded also as <br><b> https://mizugadro.mydns.jp/BOOK/468.pdf </b>(y.2020) 

<ref> 
<ref> 

−  +  https://mizugadro.mydns.jp/BOOK/466.pdf <br> 

−  +  https://mizugadro.mydns.jp/BOOK/468.pdf (update) 

−  D.Kouznetov. Superfunctions. 
+  D.Kouznetov. Superfunctions. 2020. <br> 
+  http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf (its preliminary version) <br> 

+  </ref> 

+  
+  There is also Russian version "[[Суперфункции]]" (y. 2014) 

+  <ref> 

+  https://www.morebooks.de/store/ru/book/Суперфункции/isbn/9783659562020 

+  Дмитрий Кузнецов. [[Суперфункции]]. 

+  ISBN13: 9783659562020 

+  ISBN10: 3659562025 

+  EAN: 9783659562020 

+  <br> 

+  is loaded as<br> 

+  https://mizugadro.mydns.jp/BOOK/202.pdf <br> 

+  http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf 

+  <br> 

</ref> 
</ref> 

==Covers== 
==Covers== 

+  <div class="thumb tright" style="float:right;width:194px"> 

+  <div style="width:210px"> 

+  [[File:Tetreal10bx10d.png210px]]<small> 

+  <p style="lineheight:100%"> 

+  [[Tetration]]: 

+  \(y=\mathrm{tet}_b(x)\) for various \(b\!>\!1\), Fig.17.1</p> 

+  </small> 

+  
+  [[File:Ausintay40t50.jpg210px]]<small><p style="lineheight:100%"> 

+  [[Complex mapMap]] of truncated Taylor expansion of abelsine [[AuSin]], Fig.22.2</p> 

+  </small> 

+  </div></div> 

+  <div class="thumb tright" style="float:right;width:134px"> 

+  <div style="width:150px"> 

+  
+  [[File:Oglavleni.png150px]]<small><p style="lineheight:100%"> 

+  Literature cited, Fig.21.1</p></small> 

+  
+  [[File:Bookwormbw.png150px]]<small><p style="lineheight:100%"> 

+  Literature not cited, Fig.21.3</p></small> 

+  </div></div> 

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

−  The following topics are indicated: 
+  The following topics are indicated:<br> 
−  Noninteger iterates of holomorphic functions. 
+  Noninteger iterates of holomorphic functions.<br> 
−  [[Tetration]] and other [[superfunction]]s. 
+  [[Tetration]] and other [[superfunction]]s.<br> 
−  algorithms, tables, graphics and [[complex map]]s. 
+  Formulas, algorithms, tables, graphics and [[complex map]]s. 
The back cover suggests short abstract of the Book and few notes about the Author. 
The back cover suggests short abstract of the Book and few notes about the Author. 

==About the topic== 
==About the topic== 

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

−  The Abel function (or abelfunction) is the inverse of superfunction, 
+  The Abel function (or abelfunction) is the inverse of superfunction, \(G=F^{1}\) 
−  The abelfunction is solution of the Abel equation 
+  The abelfunction is solution of the Abel equation 
−  +  \(G(T(z))=G(z)+1\) 

−  As the superfunction 
+  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 noninteger iterates. The number n of iterate can be real or even complex. In particular, for integer 
+  This expression allows to evaluate the noninteger iterates. The number \(n\) of iterate can be real or even complex. In particular, for integer \(n\), the iterates have the common meaning: <br> 
+  \(T^{1}\) is inverse function of \(T\),<br> 

−  $T^0(z)=z$,<br> 

−  +  \(T^0(z)=z\),<br> 

−  +  \(T^1(z)=T(z)\),<br> 

−  +  \(T^2(z)=T(T(z)) \),<br> 

+  \(T^3(z)=T(T(T(z))) \),<br> 

−  and so on. The group property holds: $T^m(T^n(z))=T^{m+n}(z)$ 

+  and so on. The group property holds: \(T^m(T^n(z))=T^{m+n}(z)\) 

−  The book is about evaluation of the [[superfunction]] 
+  The book is about evaluation of the [[superfunction]] \(F\), the [[abelfunction]] \(G\) and the noninteger [[iterate]]s of various transfer functions \(T\). 
+  
+  Here the number of iterate is indicated as superscript; so, \(\sin^2(z)=\sin(\sin(z))\), but never \(\sin(z)^2\).<br> 

+  This notation is borrowed from the [[Quantum mechanics]], where \(P^2(\psi)=P(P(\psi))\), but never \(P(\psi)^2\). 

−  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 $\sin(z)^2$.<br> 

−  This notation is borrowed from the [[Quantum mechanics]], where $P^2(\psi)=P(P(\psi))$, but never $P(\psi)^2$. 

==About the Book== 
==About the Book== 

−  Tools for evaluation of 
+  Tools for evaluation of [[superfunction]]s, [[abelfunction]]s and noninteger [[iterate]]s of holomorphic functions are collected. 
−  For a 
+  For a given [[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. 

+  In particular, the [[superfunction]]s of [[factorial]], exponent, sin; the holomorphic extensions of the [[logistic sequence]] and of the Ackermann functions. Among Ackermanns, [[tetration]] (mainly to the base b>1) and [[pentation]] (to base e) are presented. 

−  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. 

+  The efficient algorithm for the evaluation of [[superfunction]]s and [[abelfunction]]s are described. 

−  Superfunctions significantly extend the set of functions that can be used in scientific research and technical design. 

+  The graphics and [[complex map]]s are plotted. The possible applications are discussed. 

−  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. 

+  [[Superfunction]]s 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 arithmetic, [[Cauchi integral]] and the principles of the asymptotical analysis should help at the reading. 

==About the Author== 
==About the Author== 

Line 64:  Line 120:  
Dmitrii Kouznetsov 
Dmitrii Kouznetsov 

−  Graduated from the Physics Department of the 
+  Graduated from the Physics Department of the Moscow State University (1980). 
−  +  Work: USSR, Mexico, USA, Japan.<br> 

−  Century 20: 
+  Century 20:<br> 
−  quantum stability of the optical soliton 
+  Proven the quantum stability of the optical soliton.<br> 
−  the low bound of the quantum noise of nonlinear 
+  Suggested the low bound of the quantum noise of nonlinear amplifier.<br> 
−  +  Indicated the limit of the single mode approximation in the quantum optics. 

+  <br> 

−  approximation in the quantum optics. <br> 

Century 21: 
Century 21: 

−  Theorem about boundary 
+  Theorem about boundary behavior of modes of Dirichlet laplacian.<br> 
−  Theory of ridged atomic mirrors 
+  Theory of ridged atomic mirrors.<br> 
+  Fundamental limit of power scaling of disk lasers.<br> 

+  Formalism of superfunctions.<br> 

+  TORI axioms. 

==Summary== 
==Summary== 

Line 79:  Line 138:  
−  +  \(T\)\( ~ ~ ~ ~ ~\) [[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]] 

+  \(\displaystyle F(z)=\frac{1}{2\pi \mathrm i} \oint \frac{F(t) \, \mathrm d t}{tz}\) \(~ ~ ~\) [[Cauchi integral]] 

−  $G\big(T(z)\big)=G(z)+1$ $~ ~ ~$ [[Abel equation]], [[abelfunction]] 

+  \(\mathrm{tet}_b(z\!+\!1)=b^{\mathrm{tet}_b(z)}\) \(~ ~ ~\) [[tetration]] to base \(b\) 

−  $F\big(G(z)\big)=z$ $~ ~ ~ ~ ~$ [[Identity function]] 

+  \(\mathrm{tet}_b(0)=1\) \(~, ~ ~\) \( \mathrm{tet}_b\big(\mathrm{ate}_b(z)\big)=z\) 

−  $T^n(z)=F\big(n+G(z)\big)$ $~ ~ ~$ $n$th [[iterate]] 

+  \(\mathrm{ate}_b(b^z)=\mathrm{ate}_b(z)+1\) \(~ ~\) [[arctetration]] to base \(b\) 

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

−  +  \(\exp_b^{~n}(z)=\mathrm{tet}_b\big(n+\mathrm{ate}_b(z)\big)\) \(~ ~\) \(n\)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]] 

+  ==Recent advance== 

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

+  Most of results, presented in the book, are published in scientific journals; the links (without numbers) are supplied at the bottom. 

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

+  After the appearance of the first version of the Book (2014), certain advances are observed about evaluation of [[tetration]] 

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

+  of complex argument; the new algorithm is suggested <ref> 

+  http://journal.kkms.org/index.php/kjm/article/view/428 

+  William Paulsen. 

+  Finding the natural solution to f(f(x))=exp(x). 

+  Korean J. Math. Vol 24, No 1 (2016) pp.81106. 

+  </ref><ref> 

+  https://link.springer.com/article/10.1007/s1044401795241 

+  William Paulsen, Samuel Cowgill. 

+  Solving F(z + 1) = b ^ F(z) in the complex plane. 

+  Advances in Computational Mathematics, December 2017, Volume 43, Issue 6, pp 1261–1282 

+  </ref><ref> 

+  https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pqorigsite=gscholar&cbl=18750&diss=y 

+  Cowgill, Samuel. Exploring Tetration in the Complex Plane. 

+  Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680. 

+  </ref><ref> 

+  https://link.springer.com/article/10.1007/s1044401896157 

+  William Paulsen. 

+  Tetration for complex bases. 

+  Advances in Computational Mathematics, 2018.06.02. 

+  </ref>. These results are not included in the book. 

==References== 
==References== 

Line 190:  Line 275:  
[[Shoka function]], 
[[Shoka function]], 

[[Superfunction]], 
[[Superfunction]], 

+  [[SuperFactorial]], 

+  [[SuSin]], 

+  [[SuTra]], 

+  [[SuZex]], 

[[Tania function]], 
[[Tania function]], 

+  [[Tetration]], 

[[Trappmann function]], 
[[Trappmann function]], 

[[Tetration]], 
[[Tetration]], 
Latest revision as of 08:20, 3 May 2021
Superfunctions is book about evaluation of abelfunctions, superfunctions, and the noninteger iterates of holomorphic functions. In particular, results for tetration, arctetration and iterates of exponential are presented, ISBN 6202672862 ^{[1]}^{[2]}
https://www.morebooks.de/store/gb/book/superfunctions/isbn/9786202672863
Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
Cover at Lambert Academic Publishing, 2020
The Book is loaded also as
https://mizugadro.mydns.jp/BOOK/468.pdf (y.2020)
^{[3]}
There is also Russian version "Суперфункции" (y. 2014) ^{[4]}
Contents
Covers
At the front cover, the complex map of natural tetration is shown.
The following topics are indicated:
Noninteger 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.
About the topic
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 noninteger 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 noninteger iterates of various transfer functions \(T\).
Here the number of iterate is indicated as superscript; so, \(\sin^2(z)=\sin(\sin(z))\), but never \(\sin(z)^2\).
This notation is borrowed from the Quantum mechanics, where \(P^2(\psi)=P(P(\psi))\), but never \(P(\psi)^2\).
About the Book
Tools for evaluation of superfunctions, abelfunctions and noninteger 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 abelfuction is inverse of F. In particular, the superfunctions of factorial, exponent, sin; the holomorphic extensions of the logistic sequence and of the Ackermann functions. Among Ackermanns, 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 arithmetic, Cauchi integral and the principles of the asymptotical analysis should help at the reading.
About the Author
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 behavior of modes of Dirichlet laplacian.
Theory of ridged atomic mirrors.
Fundamental limit of power scaling of disk lasers.
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
\(\displaystyle F(z)=\frac{1}{2\pi \mathrm i} \oint \frac{F(t) \, \mathrm d t}{tz}\) \(~ ~ ~\) 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
Recent advance
Most of results, presented in the book, are published in scientific journals; the links (without numbers) are supplied at the bottom.
After the appearance of the first version of the Book (2014), certain advances are observed about evaluation of tetration of complex argument; the new algorithm is suggested ^{[5]}^{[6]}^{[7]}^{[8]}. These results are not included in the book.
References
 ↑ https://www.morebooks.de/store/gb/book/superfunctions/isbn/9786202672863 Superfunctions: Noninteger iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics Paperback – July 28, 2020 by Dmitrii Kouznetsov (Author)// Publisher : LAP LAMBERT Academic Publishing (July 28, 2020)// Language: : English// Paperback : 328 pages// ISBN10 : 6202672862// ISBN13 : 9786202672863// Item Weight : 1.18 pounds// Dimensions : 5.91 x 0.74 x 8.66 inches
 ↑
https://www.morebooks.de/store/gb/book/superfunctions/isbn/9786202672863
Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.  ↑
https://mizugadro.mydns.jp/BOOK/466.pdf
https://mizugadro.mydns.jp/BOOK/468.pdf (update) D.Kouznetov. Superfunctions. 2020.
http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf (its preliminary version)
 ↑
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/9783659562020
Дмитрий Кузнецов. Суперфункции.
ISBN13: 9783659562020
ISBN10: 3659562025
EAN: 9783659562020
is loaded as
https://mizugadro.mydns.jp/BOOK/202.pdf
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
 ↑ http://journal.kkms.org/index.php/kjm/article/view/428 William Paulsen. Finding the natural solution to f(f(x))=exp(x). Korean J. Math. Vol 24, No 1 (2016) pp.81106.
 ↑ https://link.springer.com/article/10.1007/s1044401795241 William Paulsen, Samuel Cowgill. Solving F(z + 1) = b ^ F(z) in the complex plane. Advances in Computational Mathematics, December 2017, Volume 43, Issue 6, pp 1261–1282
 ↑ https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pqorigsite=gscholar&cbl=18750&diss=y Cowgill, Samuel. Exploring Tetration in the Complex Plane. Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680.
 ↑ https://link.springer.com/article/10.1007/s1044401896157 William Paulsen. Tetration for complex bases. Advances in Computational Mathematics, 2018.06.02.
The book combines the main results from the following publications:
http://www.ams.org/mcom/200978267/S0025571809021887/home.html
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex zplane. Mathematics of Computation 78 (2009), 16471670.
http://www.jointmathematicsmeetings.org/journals/mcom/201079271/S0025571810023422/S0025571810023422.pdf
http://www.ams.org/journals/mcom/201079271/S0025571810023422/home.html
http://eretrandre.org/rb/files/Kouznetsov2009_215.pdf
http://www.ils.uec.ac.jp/~dima/PAPERS/2010q2.pdf
http://mizugadro.mydns.jp/PAPERS/2010q2.pdf
D.Kouznetsov, H.Trappmann. Portrait of the four regular superexponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.17271756.
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
http://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.612.
http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf
D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.3145.
http://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.612
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Keywords
Abel function, Book, Doya function, Iteration, Keller function, Maple and tea, LambertW, Shoka function, Superfunction, SuperFactorial, SuSin, SuTra, SuZex, Tania function, Tetration, Trappmann function, Tetration, Zex function,