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 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


is holomorphic function on , is holomorphic function on

Then and only then
is superfunction of on

Definition 2: Abel function


is superfunction on on
is holomorphic on

Then and only then

id Abel function in with respect to on .


Properties of Abel functions