Abstract :
Various properties of Jacobian elliptic functions can be put in a form that remains valid under permutation
of the first three of the letters c, d, n, and s that are used in pairs to name the functions. In most
cases 12 formulas are thereby replaced by three: one for the three names that end in s, one for the three
that begin with s, and one for the six that do not involve s. The properties thus unified in the present paper
are linear relations between squared functions (16 relations being replaced by five), differential equations,
and indefinite integrals of odd powers of a single function. In the last case the unification entails the elementary
function RC(x, y) = RF (x, y, y), where RF (x, y, z) is the symmetric elliptic integral of the first
kind. Explicit expressions in terms of RC are given for integrals of first and third powers, and alternative
expressions are given with RC replaced by inverse circular, inverse hyperbolic, or logarithmic functions.
Three recurrence relations for integrals of odd powers hold also for integrals of even powers.
Published by Elsevier Inc
