File:Sunem1map6.jpg

Complex map of function SuNem with parameter $q\!=\!1$;

$u\!+\!\mathrm i v=\mathrm{SuNem}_1(x\!+\!\mathrm i y)$

Description
Nemtsov function is polynomial of the special kind,

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

where $q$ is positive parameter.

For the Nemtsov function, its superfunction SuNem is solution $F=\mathrm{SuNem}_q$ of the transfer equation

$F(z\!+\!1)=\mathrm{Nem}_q(F(z))$

whit specific behaviour at -infinity

$\displaystyle \mathrm{SuNem}_q(z) = \frac{1}{\sqrt{-2z}} \left(1+O\left( \frac{1}{\sqrt{-2z}} \right)\right)$

and specific value at zero, $\mathrm{SuNem}_q(0)=1$.

The Nemtsov function is suggested as an example of the transfer finction, for wich the exotic iterates cannot be constructed with algorithms, presented in the First Russian version of the book Superfunctions.

Mathematica generator of the algorithm
(* Coefficients of the asymptotic expansion of function SuNem can be computed with the Mathematica code below *)

(* The resulting array of coefficients should be stored in file sunema.txt, this file should be included at the compilation of the code below *)

C++ generator of the map
// Files ado.cin, conto.cin, sune.cin, sunema.txt should be loaded in order to compile the code below