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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
MathSciNet review: 929548
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Abstract: We consider the complete hyperelliptic integral

$\displaystyle J(a) = \int_{\alpha (a)}^{\beta (a)} {\frac{{dx}}{{\sqrt {a - {P_n}(x)} }},}$

where $ a > 0$ and

$\displaystyle {P_n}(x) = \sum\limits_{k = 2}^n {{\lambda _k}{x^k},}$

with $ {\lambda _2} > 0,[\alpha ,\beta ]$ being the connected component of $ \{ x\vert{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.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1988 American Mathematical Society

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