Difference between revisions of "Superfunctions"

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[[Superfunctions]] is book about evaluation of [[abelfunction]]s, [[superfunction]]s, and the non-integer [[iterate]]s of holomorphic functions.
<div class="thumb tright" style="float:right;width:284px">
 
  +
In particular, results for [[tetration]], [[arctetration]] and [[iterate]]s of [[exponential]] are presented, ISBN 6202672862
<div style="width:300px">
 
  +
<ref>
[[File:437front.jpg|300px]]
 
  +
https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3
Front cover of the Book
 
  +
Superfunctions: Non-integer 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//
  +
ISBN-10 : 6202672862//
  +
ISBN-13 : 978-6202672863//
  +
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/978-620-2-67286-3 <br>
  +
Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
  +
</ref>
   
  +
https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3<br>
[[File:437back.jpg|300px]]
 
  +
Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
Back cover of the Book
 
  +
<div class="thumb tright" style="float:right;width:384px">
 
  +
<div style="width:400px">
[[File:Tetreal10bx10d.png|300px]]
 
  +
[[File:978-620-2-67286-3-full.jpg|400px]]
Explicit plot from the Book: [[tetration]].
 
  +
Cover at [[Lambert Academic Publishing]], 2020
\(y=\mathrm{tet}_b(x)\) versus \(x\) for various \(b\!>\!1\), Fig.17.1
 
 
[[File:Ausintay40t50.jpg|300px]]
 
[[Complex map]] from the Book: truncated Taylor expansion of abelsine [[AuSin]], Fig.22.2
 
</div></div>
 
<div class="thumb tright" style="float:right;width:134px">
 
<div style="width:150px">
 
[[File:Oglavleni.png|150px]]
 
Literature cited in the Book, Fig.21.1
 
 
[[File:Bookwormbw.png|150px]]
 
Literature not cited in the Book, Fig.21.3
 
 
</div></div>
 
</div></div>
  +
The Book is loaded also as <br><b> https://mizugadro.mydns.jp/BOOK/468.pdf </b>(y.2020)
  +
<ref>
  +
https://mizugadro.mydns.jp/BOOK/466.pdf <br>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf (update)
  +
D.Kouznetov. Superfunctions. 2020. <br>
  +
http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf (its preliminary version) <br>
  +
</ref>
   
[[Superfunctions]] is book, English remake of the Russian book [[Суперфункции]], (since 2014).
+
There is also Russian version "[[Суперфункции]]" (y. 2014)
 
<ref>
 
<ref>
 
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
 
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
Дмитрий Кузнецов. Суперфунцкии.
+
Дмитрий Кузнецов. [[Суперфункции]].
 
ISBN-13: 978-3-659-56202-0
 
ISBN-13: 978-3-659-56202-0
 
ISBN-10: 3659562025
 
ISBN-10: 3659562025
 
EAN: 9783659562020
 
EAN: 9783659562020
 
<br>
 
<br>
  +
is loaded as<br>
  +
https://mizugadro.mydns.jp/BOOK/202.pdf <br>
 
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
 
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
 
<br>
 
<br>
https://mizugadro.mydns.jp/BOOK/202.pdf
 
 
</ref>
 
</ref>
   
  +
==Covers==
The book describes evaluation of [[abelfunction]]s, [[superfunctions]], and the non-integer [[iterate]]s.
 
  +
<div class="thumb tright" style="float:right;width:194px">
In particular, results for [[tetration]], [[arctetration]] and [[iterate]]s of [[exponential]] are presented.
 
  +
<div style="width:210px">
  +
[[File:Tetreal10bx10d.png|210px]]<small>
  +
<p style="line-height:100%">
  +
[[Tetration]]:
  +
\(y=\mathrm{tet}_b(x)\) for various \(b\!>\!1\), Fig.17.1</p>
  +
</small>
   
  +
[[File:Ausintay40t50.jpg|210px]]<small><p style="line-height:100%">
The English version is loaded as https://mizugadro.mydns.jp/BOOK/444.pdf
 
  +
[[Complex map|Map]] of truncated Taylor expansion of abelsine [[AuSin]], Fig.22.2</p>
<ref>
 
  +
</small>
http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf (a little bit out of date) <br>
 
  +
</div></div>
https://mizugadro.mydns.jp/BOOK/444.pdf
 
  +
<div class="thumb tright" style="float:right;width:134px">
D.Kouznetov. Superfunctions. 2018.
 
  +
<div style="width:150px">
</ref>
 
  +
  +
[[File:Oglavleni.png|150px]]<small><p style="line-height:100%">
  +
Literature cited, Fig.21.1</p></small>
  +
  +
[[File:Bookwormbw.png|150px]]<small><p style="line-height:100%">
  +
Literature not cited, Fig.21.3</p></small>
  +
</div></div>
   
==Covers==
 
   
 
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>
Non-integer iterates of holomorphic functions.
+
Non-integer iterates of holomorphic functions.<br>
[[Tetration]] and other [[superfunction]]s. Formulas,
+
[[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.
Line 59: Line 81:
 
==About the topic==
 
==About the topic==
 
Assume some given holomorphic function \(T\).
 
Assume some given holomorphic function \(T\).
The superfunction is holomorphic solution F of equation
+
The superfunction is holomorphic solution F of equation &nbsp;
\(T(F(z))=F(z+1)\)
+
\(T(F(z))=F(z\!+\!1)\)
   
 
The Abel function (or abelfunction) is the inverse of superfunction, \(G=F^{-1}\)
 
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 &nbsp;
 
\(G(T(z))=G(z)+1\)
 
\(G(T(z))=G(z)+1\)
   
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:
+
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: &nbsp;
 
\(T^n(z)=F(n+G(z))\)
 
\(T^n(z)=F(n+G(z))\)
   
This expression allows to evaluate the non-integer iterates. The number n of iterate can be real or even complex. In particular, for integer \(n\), the iterates have the common meaning: \(T^{-1}\) is inverse function of \(T\),<br>
+
This expression allows to evaluate the non-integer iterates. The number \(n\) of iterate can be real or even complex. In particular, for integer \(n\), the iterates have the common meaning: <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>
 
\(T^1(z)=T(z)\),<br>
 
\(T^2(z)=T(T(z)) \),<br>
 
\(T^2(z)=T(T(z)) \),<br>
 
\(T^3(z)=T(T(T(z))) \),<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: &nbsp; \(T^m(T^n(z))=T^{m+n}(z)\)
   
 
The book is about evaluation of the [[superfunction]] \(F\), the [[abelfunction]] \(G\) and the non-integer [[iterate]]s of various transfer functions \(T\).
 
The book is about evaluation of the [[superfunction]] \(F\), the [[abelfunction]] \(G\) and the non-integer [[iterate]]s of various transfer functions \(T\).
   
The special notation is used in through the book; the number of iterate is indicated as superscript. For example,
+
Here the number of iterate is indicated as superscript; so,&nbsp; \(\sin^2(z)=\sin(\sin(z))\), but never \(\sin(z)^2\).<br>
In these notations, \(\sin^2(z)=\sin(\sin(z))\), but never \(\sin(z)^2\).<br>
+
This notation is borrowed from the [[Quantum mechanics]], where &nbsp; \(P^2(\psi)=P(P(\psi))\), but never \(P(\psi)^2\).
  +
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 superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected.
+
Tools for evaluation of [[superfunction]]s, [[abelfunction]]s and non-integer [[iterate]]s of holomorphic functions are collected.
For a giver transfer function T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfuction is inverse of F.
+
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 95: Line 120:
 
Dmitrii Kouznetsov
 
Dmitrii Kouznetsov
   
Graduated from the Physics Department of the
+
Graduated from the Physics Department of the Moscow State University (1980).
Moscow State University (1980). Work: USSR, Mexico, USA, Japan.<br>
+
Work: USSR, Mexico, USA, Japan.<br>
Century 20: Proven the
+
Century 20:<br>
quantum stability of the optical soliton, suggested
+
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>
amplifier, indicated the limit of the single mode
+
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 behaviour of modes of Dirichlet laplacian,
+
Theorem about boundary behavior of modes of Dirichlet laplacian.<br>
Theory of ridged atomic mirrors, formalism of superfunctions, TORI axioms.
+
Theory of ridged atomic mirrors.<br>
  +
Fundamental limit of power scaling of disk lasers.<br>
  +
Formalism of superfunctions.<br>
  +
TORI axioms.
   
 
==Summary==
 
==Summary==
Line 146: Line 174:
 
==Recent advance==
 
==Recent advance==
   
Most of results presented in the book, are published in scientific journals; the links (without numbers) are supplied at the bottom.
+
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, certain advances are observed about evaluation of [[tetration]]
+
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, that seems to be mode efficient, than the [[Cauchi integral]] described in the Book.
+
of complex argument; the new algorithm is suggested <ref>
<ref>
 
 
http://journal.kkms.org/index.php/kjm/article/view/428
 
http://journal.kkms.org/index.php/kjm/article/view/428
 
William Paulsen.
 
William Paulsen.
Line 158: Line 185:
 
https://link.springer.com/article/10.1007/s10444-017-9524-1
 
https://link.springer.com/article/10.1007/s10444-017-9524-1
 
William Paulsen, Samuel Cowgill.
 
William Paulsen, Samuel Cowgill.
Solving F(z + 1) = b F(z) in the complex plane.
+
Solving F(z + 1) = b ^ F(z) in the complex plane.
 
Advances in Computational Mathematics, December 2017, Volume 43, Issue 6, pp 1261–1282
 
Advances in Computational Mathematics, December 2017, Volume 43, Issue 6, pp 1261–1282
 
</ref><ref>
 
</ref><ref>
Line 169: Line 196:
 
Tetration for complex bases.
 
Tetration for complex bases.
 
Advances in Computational Mathematics, 2018.06.02.
 
Advances in Computational Mathematics, 2018.06.02.
  +
</ref>. These results are not included in the book.
</ref>
 
   
 
==References==
 
==References==

Revision as of 08:20, 3 May 2021

Superfunctions is book about evaluation of abelfunctions, superfunctions, and the non-integer 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/978-620-2-67286-3
Dmitrii Kouznetsov. Superfunctions. 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]

Covers

Tetreal10bx10d.png

Tetration: \(y=\mathrm{tet}_b(x)\) for various \(b\!>\!1\), Fig.17.1

Ausintay40t50.jpg

Map of truncated Taylor expansion of abelsine AuSin, Fig.22.2

Oglavleni.png

Literature cited, Fig.21.1

Bookwormbw.png

Literature not cited, Fig.21.3


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

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

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

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

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

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 non-integer iterates of holomorphic functions are collected. For a given transfer function T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The 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

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

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

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

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

\(\exp_b^{~n}(z)=\mathrm{tet}_b\big(n+\mathrm{ate}_b(z)\big)\) \(~ ~\) \(n\)th iterate of function \(~\) \(z\!\mapsto\! b^z\)

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

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

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

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

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

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

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

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

  1. https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3 Superfunctions: Non-integer 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// ISBN-10 : 6202672862// ISBN-13 : 978-6202672863// Item Weight : 1.18 pounds// Dimensions : 5.91 x 0.74 x 8.66 inches
  2. https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3
    Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
  3. 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)
  4. https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 Дмитрий Кузнецов. Суперфункции. ISBN-13: 978-3-659-56202-0 ISBN-10: 3659562025 EAN: 9783659562020
    is loaded as
    https://mizugadro.mydns.jp/BOOK/202.pdf
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
  5. 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.81-106.
  6. https://link.springer.com/article/10.1007/s10444-017-9524-1 William Paulsen, Samuel Cowgill. Solving F(z + 1) = b ^ F(z) in the complex plane. Advances in Computational Mathematics, December 2017, Volume 43, Issue 6, pp 1261–1282
  7. https://search.proquest.com/openview/cb7af40083915e275005ffca4bfd4685/1?pq-origsite=gscholar&cbl=18750&diss=y Cowgill, Samuel. Exploring Tetration in the Complex Plane. Arkansas State University, ProQuest Dissertations Publishing, 2017. 10263680.
  8. https://link.springer.com/article/10.1007/s10444-018-9615-7 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/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 78 (2009), 1647-1670.

http://www.jointmathematicsmeetings.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/S0025-5718-10-02342-2.pdf
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
http://eretrandre.org/rb/files/Kouznetsov2009_215.pdf
http://www.ils.uec.ac.jp/~dima/PAPERS/2010q2.pdf
http://mizugadro.mydns.jp/PAPERS/2010q2.pdf
D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

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.6-12.

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.31-45.

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.6-12

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

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

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

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

http://www.scirp.org/journal/PaperInformation.aspx?PaperID=36560
http://mizugadro.mydns.jp/PAPERS/2013jmp.pdf
D.Kouznetsov. TORI axioms and the applications in physics. Journal of Modern Physics, 2013, v.4, p.1151-1156.

http://www.ingentaconnect.com/content/asp/asl/2013/00000019/00000003/art00071 http://mizugadro.mydns.jp/PAPERS/2012thaiSuper.pdf D.Kouznetsov. Recovery of Properties of a Material from Transfer Function of a Bulk Sample (Theory). Advanced Science Letters, Volume 19, Number 3, March 2013, pp. 1035-1038(4).

http://link.springer.com/article/10.1007/s10043-013-0058-6 D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.

http://www.m-hikari.com/ams/ams-2013/ams-129-132-2013/kouznetsovAMS129-132-2013.pdf http://mizugadro.mydns.jp/PAPERS/2013hikari.pdf D.Kouznetsov. Entire function with logarithmic asymptotic. Applied Mathematical Sciences, 2013, v.7, No.131, p.6527-6541.

http://www.pphmj.com/references/8246.htm http://mizugadro.mydns.jp/PAPERS/2014susin.pdf D.Kouznetsov. Super sin. Far East Jourmal of Mathematical Science, v.85, No.2, 2014, pages 219-238.

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,