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Abel function
From Citizendium, the Citizens' Compendium
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.
Contents |
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 .