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 m1 (u, k) ds m2 (u, k) ns m3 (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.