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

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 necesary 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 .

Examples

Properties of Abel functions

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.