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Incomplete hyperelliptic integrals and hypergeometric series


Authors: J.-F. Loiseau, J.-P. Codaccioni and R. Caboz
Journal: Math. Comp. 53 (1989), 335-342
MSC: Primary 33A35
DOI: https://doi.org/10.1090/S0025-5718-1989-0972371-2
MathSciNet review: 972371
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Abstract: We consider the incomplete hyperelliptic integral \[ H(a,X) = \int _0^X {\frac {{dx}}{{\sqrt {a - {\lambda _2}{x^2} - {\lambda _n}{x^n}} }}} \] with $a > 0$, ${\lambda _2} > 0$, $n > 2$, where X belongs to the connected component of $\{ x|{\lambda _2}{x^2} + {\lambda _n}{x^n} < a\}$ containing the origin. Continuing previous work on the complete hyperelliptic integral, we express in this paper $H(a,X)$ as a convergent series of hypergeometric type. A brief survey of some applications to algebraic equations and mechanics is then given.


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Article copyright: © Copyright 1989 American Mathematical Society