Asymptotics of the Gauss Hypergeometric Function with Large Parameters, I
نویسنده
چکیده
We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ ε1λ ,b+ ε2λ ;c+ ε3λ ;z) as |λ | →∞ when the ε j are finite by an application of the method of steepest descents, thereby extending previous results corresponding to ε j = 0, ±1 . By means of connection formulas satisfied by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0 , ε2 = 0 , ε3 = 1 and (ii) ε1 > 0 , ε2 = −1 , ε3 = 0 ; the third case ε1, ε2 > 0 , ε3 = 1 is deferred to Part II. The resulting expansions are of Poincaré type and hold in restricted domains of the complex z -plane. Numerical results illustrating the accuracy of the different expansions are given. Mathematics subject classification (2010): Primary 33C05, 34E05, 41A60.
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