Cyclic Identities Involving Jacobi Elliptic Functions

Details Cyclic Identities Involving Jacobi Elliptic Functions Cyclic Identities Involving Jacobi Elliptic Functions

2002-01-03

arxiv.org/abs/math-ph/0201004v2

Physics

Mathematical Physics

14 pages, 0 figures

Scientific paper

10.1063/1.1484541

We state and discuss numerous mathematical identities involving Jacobi elliptic functions sn(x,m), cn(x,m), dn(x,m), where m is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either 2K(m)/p or 4K(m)/p, where p is an integer and K(m) is the complete elliptic integral of the first kind. Each p-point identity of rank r involves a cyclic homogeneous polynomial of degree r (in Jacobi elliptic functions with p equally spaced arguments) related to other cyclic homogeneous polynomials of degree r-2 or smaller. Identities corresponding to small values of p,r are readily established algebraically using standard properties of Jacobi elliptic functions, whereas identities with higher values of p,r are easily verified numerically using advanced mathematical software packages.

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