Ueber die Moeglichkeiten einer schlichten konformen. Abbildung des Einheitskreises durch die hypergeometrische Funktion von Gauss

Abstract

1. In this report we find the closed discs with center z = 0, which the hypergeometric function of Gauss (1) and its derivatives (2) mapp univalent conformal. Investigated is the case, when the point z = 1 of the cut (1, + ∞) can be added toward the closed unit disc | z | ≤ 1 , which is mapped conformai from the respective hypergeometric functions. 2. We treat the afiliation of the hypergeometric function to the class C functions of Caratheodory, in the closed unit disc | z | ≤ 1, in which it is univalent. More exactly we introduce the following definitions : 1) We name positive univalent (or function CS), every analitic function, when is C function of Caratheodory and maps the closed unit disc | z | ≤ 1 in the univalent domain. 2) We name normal every positive univalent function (every CS function) when the closed univalent unit disc | z | ≤ 1 is the biggest. In our research we build the class normal hypergeometric functions in the treated aspect. 3. The received results are applied for the full eleptic integrals of Legendre of first and second kin. We have shown the way for the research of the set problems and specially for the bordering univalent conformai mapping in the mathematic and theoretic physics for the related with the hypergeometric function special functions.