Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions

Abstract

Any product of real powers of Jacobian elliptic functions can be written in the form cs m 1 (u, k) ds m 2 (u, k) ns m 3 (u, k). If all three m's are even integers, the indefinite integral of this product with respect to u is a constant times a multivariate hypergeometric function R −a (b 1 , b 2 , b 3 ; x, y, z) with half-odd-integral b's and −a + b 1 + b 2 + b 3 = 1, showing it to be an incomplete elliptic integral of the second kind unless all three m's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables {x, y, z} = {cs 2 , ds 2 , ns 2 }, allowing as many as six integrals to take a unified form. Thirty R-functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting x, y, and z in R D (x, y, z) = R −3/2 (1 2 , 1 2 , 3 2 ; x, y, z), which is symmetric in its first two variables and has an efficient algorithm for numerical computation.