Abel function

For some given Transfer function $T$, the Abel function $G$ is inverse function of the corresponding superfunction $F$, id est, $G=F^{-1}$.

The Abel equation relates the Abel function $G$ and the transfer function $T$:

$G(T(z))=G(z)+1$

In certain range of values of $z$, this equation is equivalent of the Transfer equation

$T(F(z))=F(z\!+\!1)$

The transfer function $T$ is supposed to be known; then, the problem is to find the corresponding superfunction(s) and/or the Abel function(s).

The examples of the transfer functions, the superfunctions and the [[Abel functoons[[ $G$ are suggested in the Table of superfunctions.

Superfunction and iterates of the transfer function

The superfunction and the Abel function allow to define the $n$th iteration of the corresponding transfer function $T$ in the following form:

$T^n(z)=F(n+G(z))$

This expression may hold for wide range of values of $z$ and $n$ from the set of complex numbers. In particular, for integer values of $n$,

$T^{-1}$ is inverse function of $T$

$T^0(z)=z$,

$T^1(z)=T(z)$

$T^2(z)=T(T(z))$

and so on. The non-integer iteration of function allows to express such functions as square root of factorial and square root of exponential in terms of the superfunction and the Abel function.

Existence and unuqueness

In many cases, the superfunction $F$ can be constructed with the regular iteration; then, for given superfunction, $G$ is unique. However, the regular iteration can be realized at various fixed points of the transfer function $T$ (if it has many fixed points). Then, hte superfunctions constructed with regular iteration, are different; in particular, they may have different periodicity. Sequently, the Abel functions are also different.

In order to define the unique Abel function $G$, the additional requirements on its asymptotic behavior should be applied .