Generalized Landen Transformation Formulas for Jacobi Elliptic Functions

Abstract

Landen transformation formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by changing integration variables in elliptic integrals.We rediscover known results as well as obtain more generalized Landen formulas from a very different perspective, by making use of the recently obtained periodic solutions of physically interesting nonlinear differential equations and numerous remarkable new cyclic identities involving Jacobi elliptic functions. We find that several of our Landen transformations have a rather different and substantially more elegant appearance compared to the forms usually found in the literature. Further, by making use of the cyclic identities discovered recently, we also obtain some entirely new sets of Landen transformations. This paper is an expanded and revised version of our previous paper math-ph/0204054.