# Abel function

(Redirected from Abelfunction)

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.

## Etymology

The Abel function and the Abel Equation are named after Neils Henryk Abel .

## 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 .