Asymptotics of the Gauss hypergeometric function with large parameters, II

Richard B. Paris

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Abstract

We obtain asymptotic expansions by application of the method of steepest descents for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+λ;z) as |λ| → ∞ when 0<ε1 <1 and ε1 >1 where, without loss of generality, it is supposed that ε1 6ε2 . The resulting expansions are of Poincar´e type and break down in the neighbourhood of certain critical points in the z-plane. Numerical results illustrating the accuracy of the different expansions are given.
Original languageEnglish
Pages (from-to)1-15
Number of pages15
JournalJournal of Classical Analysis
Volume3
Issue number1
Publication statusPublished - 2013

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Gauss Hypergeometric Function
Steepest Descent
Poincaré
Breakdown
Asymptotic Expansion
Critical point
Numerical Results

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abstract = "We obtain asymptotic expansions by application of the method of steepest descents for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+λ;z) as |λ| → ∞ when 0<ε1 <1 and ε1 >1 where, without loss of generality, it is supposed that ε1 6ε2 . The resulting expansions are of Poincar´e type and break down in the neighbourhood of certain critical points in the z-plane. Numerical results illustrating the accuracy of the different expansions are given.",
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Asymptotics of the Gauss hypergeometric function with large parameters, II. / Paris, Richard B.

In: Journal of Classical Analysis, Vol. 3, No. 1, 2013, p. 1-15.

Research output: Contribution to journalArticle

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N2 - We obtain asymptotic expansions by application of the method of steepest descents for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+λ;z) as |λ| → ∞ when 0<ε1 <1 and ε1 >1 where, without loss of generality, it is supposed that ε1 6ε2 . The resulting expansions are of Poincar´e type and break down in the neighbourhood of certain critical points in the z-plane. Numerical results illustrating the accuracy of the different expansions are given.

AB - We obtain asymptotic expansions by application of the method of steepest descents for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+λ;z) as |λ| → ∞ when 0<ε1 <1 and ε1 >1 where, without loss of generality, it is supposed that ε1 6ε2 . The resulting expansions are of Poincar´e type and break down in the neighbourhood of certain critical points in the z-plane. Numerical results illustrating the accuracy of the different expansions are given.

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