Asymptotics of the Gauss hypergeometric function with large parameters, I

Richard B. Paris

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Abstract

We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are finite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 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 Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.
Original languageEnglish
Pages (from-to)183-203
Number of pages21
JournalJournal of Classical Analysis
Volume2
Issue number2
DOIs
Publication statusPublished - 2013

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Gauss Hypergeometric Function
Steepest Descent
Hypergeometric Functions
Poincaré
Asymptotic Expansion
Numerical Results

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Paris, Richard B. / Asymptotics of the Gauss hypergeometric function with large parameters, I. In: Journal of Classical Analysis. 2013 ; Vol. 2, No. 2. pp. 183-203.
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Paris, RB 2013, 'Asymptotics of the Gauss hypergeometric function with large parameters, I', Journal of Classical Analysis, vol. 2, no. 2, pp. 183-203. https://doi.org/dx.doi.org/10.7153/jca-02-15

Asymptotics of the Gauss hypergeometric function with large parameters, I. / Paris, Richard B.

In: Journal of Classical Analysis, Vol. 2, No. 2, 2013, p. 183-203.

Research output: Contribution to journal › Article

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N2 - We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are finite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 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 Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.

AB - We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are finite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 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 Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.

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Paris RB. Asymptotics of the Gauss hypergeometric function with large parameters, I. Journal of Classical Analysis. 2013;2(2):183-203. https://doi.org/dx.doi.org/10.7153/jca-02-15

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