# Abel function

Jump to navigation
Jump to search

*The content on this page originated on Wikipedia and is yet to be significantly improved. Contributors are invited to replace and add material to make this an original article.*

**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

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

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

- ,
- is holomorphic function on , is holomorphic function on

Then and only then

is
superfunction of on

### Definition 2: Abel function

If

- is superfunction on on
- ,
- is holomorphic on

Then and only then

- id Abel function in with respect to on .