### 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 language | English |
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Pages (from-to) | 183-203 |

Number of pages | 21 |

Journal | Journal of Classical Analysis |

Volume | 2 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 |

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## Cite this

Paris, R. B. (2013). Asymptotics of the Gauss hypergeometric function with large parameters, I.

*Journal of Classical Analysis*,*2*(2), 183-203. https://doi.org/dx.doi.org/10.7153/jca-02-15