Abel function: Difference between revisions

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imported>Eric Evers
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imported>Dmitrii Kouznetsov
Line 36: Line 36:
If
If
:<math> f </math> is <math>u,v</math> superfunction on <math>F</math> on <math>D</math>
:<math> f </math> is <math>u,v</math> superfunction on <math>F</math> on <math>D</math>
: <math> H \subseteq \mathbb{C},  <math> D \subseteq \mathbb{C},   
:<math> H \subseteq \mathbb{C}</maht>,  <math> D \subseteq \mathbb{C},</math>  
:<math>g</math> is holomorphic on <math>H</math>
:<math>g(H)\subseteq D</math>
:<math>f(g(z))=z \forall z \in H </math>
:<math>g(u)=v, ~ u\in G</math>
Then and only then
:<math> g </math> id <math> u,v</math> Abel function in <math>F</math> with respect to <math>f</math> on <math>D</math>.


==Examples==
==Examples==
==Properties of Abel funcitons==
<references/>
<references/>

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Abel function is special kind of solutions of the Abel equations used to classify then as superfunctions, and formulate conditions of the 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 iteraitonal equation

which is also called "Abel equation". There is deduction at wikipedia that show some eqiovalence of these equaitons.

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 funcitons

If

,
is holomorphic function on , is holomorphic function on

Then and only then
is superfunction of on


If

is superfunction on on
is holomorphic on

Then and only then

id Abel function in with respect to on .

Examples

Properties of Abel funcitons

  1. N.H.Abel. Determinaiton 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, p.85-100