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 $\textrm {cs}^{m_1}(u,k) \textrm {ds}^{m_2}(u,k) \textrm {ns}^{m_3}(u,k)$. 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 $\{x,y,z\} =\{\textrm {cs}^2,\textrm {ds}^2,\textrm {ns}^2$}, 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 $R_D(x,y,z) =R_{-3/2}(\frac {1}{2},\frac {1}{2}, \frac {3}{2}; x,y,z)$, which is symmetric in its first two variables and has an efficient algorithm for numerical computation.