Abel function

In mathematics 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 is class of equations which can be written in the form

[math]\displaystyle{ g(f(z))=g(z)+1 }[/math]

where function ƒ is supposed to be given, and function g is expected to be found. This equation is closely related to the iterative equation

[math]\displaystyle{ H(F(z))=F(z+1) \, }[/math]

[math]\displaystyle{ f(u)=v \, }[/math]

which is also called "Abel equation".

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

Superfunctions and Abel functions

Definition 1: Superfunction

If

[math]\displaystyle{ C \subseteq \mathbb{C} }[/math], [math]\displaystyle{ D \subseteq \mathbb{C} }[/math]

[math]\displaystyle{ F }[/math] is holomorphic function on [math]\displaystyle{ C }[/math], [math]\displaystyle{ f }[/math] is holomorphic function on [math]\displaystyle{ D }[/math]

[math]\displaystyle{ F(D) \subseteq C }[/math]

[math]\displaystyle{ f(u)=v }[/math]

[math]\displaystyle{ F(z+1)=F(f(z)) ~\forall z \in D : z\!+\!1 \in D }[/math]

Then and only then
[math]\displaystyle{ f }[/math] is [math]\displaystyle{ u,v }[/math] superfunction of [math]\displaystyle{ F }[/math] on [math]\displaystyle{ D }[/math]

Definition 2: Abel function

If

[math]\displaystyle{ f }[/math] is [math]\displaystyle{ u,v }[/math] superfunction on [math]\displaystyle{ F }[/math] on [math]\displaystyle{ D }[/math]

[math]\displaystyle{ H \subseteq \mathbb{C} }[/math], [math]\displaystyle{ D \subseteq \mathbb{C}, }[/math]

[math]\displaystyle{ g }[/math] is holomorphic on [math]\displaystyle{ H }[/math]

[math]\displaystyle{ g(H)\subseteq D }[/math]

[math]\displaystyle{ f(g(z))=z \forall z \in H }[/math]

[math]\displaystyle{ g(u)=v, ~ u\in G }[/math]

Then and only then

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