Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic expansions of the Appell's function $ F_1$

Authors: Chelo Ferreira and José L. López
Journal: Quart. Appl. Math. 62 (2004), 235-257
MSC: Primary 41A60; Secondary 33C65, 65D20
DOI: https://doi.org/10.1090/qam/2054598
MathSciNet review: MR2054598
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Abstract: The first Appell's hypergeometric function $ {F_1}\left( a, b, c, d; x, y \right)$ is considered for large values of its variables $ x$ and/or $ y$. An integral representation of $ {F_1}\left( a, b, c, d; x, y \right)$ is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of $ 1/\left( 1 - x \right)$ and/or $ 1/\left( 1 - y \right)$. For certain values of the parameters $ a, b, c$ and $ d$, two of these expansions also involve logarithmic terms in the asymptotic variables $ 1 - x$ and/or $ 1 - y$. Coefficients of these expansions are given in terms of the Gauss hypergeometric function $ _2{F_1}\left( \alpha , \beta , \gamma ; x \right)$ and its derivative with respect to the parameter $ \alpha $. All of the expansions are accompanied by error bounds for the remainder at any order of the approximation. These error bounds are obtained from the error test and, as numerical experiments show, they are considerably accurate.

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DOI: https://doi.org/10.1090/qam/2054598
Article copyright: © Copyright 2004 American Mathematical Society

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