On a functional equation for Jacobi's elliptic function cn(z;k)
type: Item
instance of: scholarly article
title: On a functional equation for Jacobi's elliptic function cn(z;k)
zbMATH Open document ID: 0546.33001
DOI: 10.1007/BF02192680
published in: Aequationes Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137033
review text: 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.
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 39B99
zbMATH DE Number: 3869727
zbMATH Keywords: cosine functional equation
zbMATH Keywords: Jacobi's elliptic cosine function
author: Hiroshi Haruki
MaRDI profile type: Publication
cites work: Q5514010
cites work: An application of Nevanlinna-pólya theorem to a cosine functional equation
cites work: Q5570370
cites work: The Functional Equation f(xy) + f(xy -1 ) = 2f(x)f(y) for Groups
cites work: Q5720427
OpenAlex ID: W2087322588
description: scientific article; zbMATH DE number 3869727