Incomplete hyperelliptic integrals and hypergeometric series

Abstract

We consider the incomplete hyperelliptic integral \[ H ( a , X ) = ∫ 0 X d x a − λ 2 x 2 − λ n x n H(a,X) = \int _0^X {\frac {{dx}}{{\sqrt {a - {\lambda _2}{x^2} - {\lambda _n}{x^n}} }}} \] with a > 0 a > 0 , λ 2 > 0 {\lambda _2} > 0 , n > 2 n > 2 , where X belongs to the connected component of { x | λ 2 x 2 + λ n x n > a } \{ 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 ) H(a,X) as a convergent series of hypergeometric type. A brief survey of some applications to algebraic equations and mechanics is then given.