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Hyperelliptic integrals and multiple hypergeometric series


Authors: Jean-Francis Loiseau, Jean-Pierre Codaccioni and Régis Caboz
Journal: Math. Comp. 50 (1988), 501-512
MSC: Primary 33A25; Secondary 33A35
DOI: https://doi.org/10.1090/S0025-5718-1988-0929548-0
MathSciNet review: 929548
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Abstract: We consider the complete hyperelliptic integral \[ J(a) = \int _{\alpha (a)}^{\beta (a)} {\frac {{dx}}{{\sqrt {a - {P_n}(x)} }},}\] where $a > 0$ and \[ {P_n}(x) = \sum \limits _{k = 2}^n {{\lambda _k}{x^k},}\] with ${\lambda _2} > 0,[\alpha ,\beta ]$ being the connected component of $\{ x|{P_n}(x) \leq a\}$ containing the origin. Using a recent result concerning the Taylor expansion of the $\delta$-Dirac function, we write $J(a)$ as a power series of a parameter involving a and the ${\lambda _k}$’s. We prove this series to be a sum of multiple hypergeometric series which reduces to a single term when the number of odd monomial terms in ${P_n}$ is less than or equal to one. The region of convergence is then studied and a few particular cases are detailed.


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Article copyright: © Copyright 1988 American Mathematical Society