A Generalization of Landen’s Quadratic Transformation Formulas for Jacobi Elliptic Functions

Landen formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by making a suitable quadratic transformation of variables in elliptic integrals. We obtain and discuss significant generalizations of the celebrated Landen formulas. Our approach is based on some recently obtained periodic solutions of physically interesting nonlinear differential equations and numerous remarkable new cyclic identities involving Jacobi elliptic functions. [email protected] [email protected] 1 Jacobi elliptic functions sn(x,m), cn(x,m) and dn(x,m), with elliptic modulus parameter m ≡ k (0 ≤ m ≤ 1) play an important role in describing periodic solutions of many linear and nonlinear differential equations of interest in diverse branches of engineering, physics and mathematics. The Jacobi elliptic functions are often defined with the help of the elliptic integral