Jacobian elliptic functions as inverses of an integral

Abstract

The 12 Jacobian elliptic functions are traditionally shown as inverses of 12 elliptic integrals, all of them being special cases of ∫yx[(a1 + b1t2)(a2 + b2t2)-1/2 dt in which all quantities are real and either y = 0 or x = ∞ or a1 + b1y2 = 0 or a1 + b1x2 = 0. A new unified treatment shows that for each of these four cases the other limit of integration is determined as the inverse function of the integral by the two products a1b2 and a2b1. Inequalities and equalities between these two and 0 distinguish the 12 Jacobian functions, the six circular functions, and the six hyperbolic functions. The proof comes from a corollary of a reduction theorem for the symmetric elliptic integral of the first kind.