Incomplete Hyperelliptic Integrals and Hypergeometric Series

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.