# Difference between revisions of "Superfunctions"

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

Superfunctions is book, English version of the Russian book Суперфункции, (since 2015). The book describes evaluation of abelfunctions, superfunctions, and the non-integer iterates. In particular, results for tetration, arctetration and iterates of exponential are presented.

## Covers

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

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

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

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

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

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

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

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

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

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

Tools for evaluation of superfunctions, abelfunctions and non-integer iterates of holomorphic functions are collected. For a giver transfer function T, the superfunction is solution F of the transfer equation F(z+1)=T(F(z)) . The abelfuction is inverse of F. In particular, thesuperfunctions of factorial, exponent, sin; the holomorphic extensions of the logistic sequence and of the Ackermann functions are suggested. from ackermanns, the tetration (mainly to the base b>1) and pentation (to base e) are presented. The efficient algorithm for the evaluation of superfunctions and abelfunctions are described. The graphics and complex maps are plotted. The possible applications are discussed. Superfunctions significantly extend the set of functions that can be used in scientific research and technical design. Generators of figures are loaded to the site TORI, http://mizugadro.mydns.jp for the free downloading. With these generators, the Readers can reproduce (and modify) the figures from the Book. The Book is intended to be applied and popular. I try to avoid the complicated formulas, but some basic knowledge of the complex arithmetics, Cauchi integral and the principles of the asymptotical analysis should help at the reading.

Dmitrii Kouznetsov

Graduated from the Physics Department of the Moscow State University (1980). Work: USSR, Mexico, USA, Japan.
Century 20: Proven the quantum stability of the optical soliton, suggested the low bound of the quantum noise of nonlinear amplifier, indicated the limit of the single mode approximation in the quantum optics.
Century 21: Theorem about boundary behaviour of modes of Dirichlet laplacian, Theory of ridged atomic mirrors, formalism of superfunctions, TORI axioms.

## Summary

The summary suggests main notations used in the Book:

$T$$~ ~ ~ ~ ~$ Transfer function

$T\big(F(z)\big)=F(z\!+\!1)$ $~ ~ ~$ Transfer equation, superfunction

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

After the appearance of the first version of the Book, 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. [2][3][4][5]

## References

1. http://www.ils.uec.ac.jp/~dima/BOOK/443.pdf (a little bit out of date)
2. 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.
3. 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
4. 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.
5. 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
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation 78 (2009), 1647-1670.

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

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

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

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

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

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

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

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

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