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ASYMPTOTICS OF A GAUSS HYPERGEOMETRIC FUNCTION WITH TWO LARGE PARAMETERS: A NEW CASE

Part of: Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  10 December 2019

J. F. HARPER Open the ORCID record for J. F. HARPER [Opens in a new window]
J. F. HARPER*
Affiliation:
School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington6140, New Zealand; e-mail: john.harper@vuw.ac.nz
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Abstract

Asymptotic expansions of the Gauss hypergeometric function with large parameters, $F(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}_{1}\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D716}_{2}\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D716}_{3}\unicode[STIX]{x1D70F};z)$ as $|\unicode[STIX]{x1D70F}|\rightarrow \infty$, are known for many special cases, but not for one that the author encountered in recent work on fluid mechanics: $\unicode[STIX]{x1D716}_{2}=0$ and $\unicode[STIX]{x1D716}_{3}=\unicode[STIX]{x1D716}_{1}z$. This paper gives the leading term for that case if $\unicode[STIX]{x1D6FD}$ is not a negative integer and $z$ is not on the branch cut $[1,\infty )$, and it shows how subsequent terms can be found.

Keywords

asymptotic expansionshypergeometric functionslarge parameters

MSC classification

Primary: 30E15: Asymptotic representations in the complex domain
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 4 , October 2020 , pp. 446 - 452
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Copyright
© Australian Mathematical Society 2019

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