# Abel function

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Abel function is a special kind of solution of the Abel equations, used to classify them as superfunctions, and formulate conditions of uniqueness.

The Abel equation [1] [2] is class of equations which can be written in the form

${\displaystyle g(f(z))=g(z)+1}$

where function ${\displaystyle f}$ is supposed to be given, and function ${\displaystyle g}$ is expected to be found. This equation is closely related to the iterational equation

${\displaystyle H(F(z))=F(z+1)}$
${\displaystyle f(u)=v}$

which is also called "Abel equation".

In general the Abel equation may have many solutions, and the additional requirements are necesary to select the only one among them.

## superfunctions and Abel functions

### Definition 1: Superfunction

If

${\displaystyle C\subseteq \mathbb {C} }$, ${\displaystyle D\subseteq \mathbb {C} }$
${\displaystyle F}$ is holomorphic function on ${\displaystyle C}$, ${\displaystyle f}$ is holomorphic function on ${\displaystyle D}$
${\displaystyle F(D)\subseteq C}$
${\displaystyle f(u)=v}$
${\displaystyle F(z+1)=F(f(z))~\forall z\in D:z\!+\!1\in D}$

Then and only then
${\displaystyle f}$ is ${\displaystyle u,v}$ superfunction of ${\displaystyle F}$ on ${\displaystyle D}$

### Definition 2: Abel function

If

${\displaystyle f}$ is ${\displaystyle u,v}$ superfunction on ${\displaystyle F}$ on ${\displaystyle D}$
${\displaystyle H\subseteq \mathbb {C} }$, ${\displaystyle D\subseteq \mathbb {C} ,}$
${\displaystyle g}$ is holomorphic on ${\displaystyle H}$
${\displaystyle g(H)\subseteq D}$
${\displaystyle f(g(z))=z\forall z\in H}$
${\displaystyle g(u)=v,~u\in G}$

Then and only then

${\displaystyle g}$ id ${\displaystyle u,v}$ Abel function in ${\displaystyle F}$ with respect to ${\displaystyle f}$ on ${\displaystyle D}$.

## References

1. N.H.Abel. Determination d'une function au moyen d'une equation qui ne contien qu'une seule variable. Oeuvres completes, Christiania, 1881.
2. G.Szekeres. "Abel's equation and regular gtowth: Variations of a theme by Abel". Experimental mathematics 7 (2): 85-100.