On a functional equation for Jacobi's elliptic function cn(z;k)
The author considers the functional equation \((1)\quad (f(x+y)+f(x- y))(f(x)^ 2+f(y)^ 2)=2f(x)f(y)(f(x+y)f(x-y)+1),\) where f:\({\mathbb{C}}\to {\mathbb{C}}\) and x,y are complex variables. He shows first that (1) is a generalization of the cosine functional equation \(f(x+y)+f(x- y)=2f(x)f(y).\) The main result proved in this paper is the following theorem: Theorem. If f is a meromorphic function of a complex variable z for \(| z| <+\infty,\) then the only solutions of (1) are \(f(z)\equiv 0\) and \(f(z)=cn(\alpha z;k)\) where \(\alpha\),k are arbitrary complex constants. Here cn denotes Jacobi's elliptic cosine function.
scientific article; zbMATH DE number 3869727