Asymptotics of a Gauss hypergeometric function with large parameters, III: Application to the Legendre functions of large imaginary order and real degree

2016/09/27 by R. B. Paris, Paris, R. B.
Mathematics · #33C05 #34E05 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C05 #msc:34E05 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1609.08365

14 pages, 4 figures

arxiv created 2016/09/27 · arxiv updated 2016/09/28

Abstract

We obtain the asymptotic expansion for the Gauss hypergeometric function F(a-λ,b+λ;c+iαλ;z) for λ→+∞ with a, b and c finite parameters by application of the method of steepest descents. The quantity α is real, so that the denominatorial parameter is complex and z is a finite complex variable restricted to lie in the sector |arg (1-z)|<π. We concentrate on the particular case a=0, b=c=1, which is associated with the Legendre functions of real degree and imaginary order. The resulting expansions are of Poincaré type and hold in restricted domains of the z-plane. An expansion is given at the coalescence of two saddle points. Numerical results illustrating the accuracy of the different expansions are given.

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