A functional equation and Jacobian elliptic functions

Abstract

The functional equation¶¶$ {f(x+y) \over f(x-y)} = {g(x) + g(y) \over g(x)-g(y)}, $¶where f and g are meromorphic functions, was encountered by P. McGill in work on Brownian motion. We solve this functional equation by a different method in the following two situations:¶1. f and g are arbitrary meromorphic functions;¶2. f and g are meromorphic functions that are real valued on the real axis.¶In the first case it is shown that there is essentially only one solution, namely f (x) = sn x, g (x) = sn x / (cn x cn x), where sn, cn and dn are Jacobian elliptic functions.