Кузнецов Дмитрий Юрьевич: другие произведения.

Superfunctions

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  • Аннотация:
    For a given holomorphic function T, superfunction F is solution of the transfer equation T(F(z))=F(z+1)

"Superfunctions" is book that is published in 2020 by the Lambert Academic Publishing (LAP). You may support the publisher, buying it at
https://www.morebooks.de/store/gb/book/superfunctions/isbn/978-620-2-67286-3

Also, you may read it online at http://mizugadro.mydns.jp/BOOK/466.pdf.
It is free. (Haliava, Sir!)

Below I describe the main topic of the Book.

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 nth iterate of transfer function T can be expressed as follows:
Tn(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,
T0(z)=z
T1(z)=T(z)
T2(z)=T(T(z))
T3(z)=T(T(T(z)))
and so on. For complex m and n, the group property holds:
Tm(Tn(z))=Tm+n(z)

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

Any critics should be appreciated.
It is difficult to type formulas here (no latex commands seem to be supported), so, I suggest
https://mizugadro.mydns.jp/t/index.php/Superfunctions
and http://mizugadro.mydns.jp/BOOK/466.pdf

The Russian version is also available at my site,
http://mizugadro.mydns.jp/BOOK/202.pdf
Since century 20, the KGB agents, performing their total war against the Human Civilization, force the Russian providers to sabotage, they make problems at reading of content of scientific sites. If read from Russia, then, use of some anonimizer or some satellite internet (independent from KGB and Roskomnadzor) may be necessary.
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