Note on the Jacobi elliptic functions
type: Item
instance of: scholarly article
title: Note on the Jacobi elliptic functions
zbMATH Open document ID: 0914.33014
author: Hyo Chul Myung
published in: Journal of Algebra
publication date: 1999-01-19T00:00:00Z
review text: The Jacobi elliptic functions \(x_1(t)=\text{sn } t\), \(x_2(t)=\text{cn } t\), \(x_3(t)=\text{dn } t\) constitute the solution of the autonomous system of quadratic differential equations \(x'(t)+x(t)^2=0\), with \(x(t)=(x_1(t),x_2(t),x_3(t))\), initial values \(x(0)=(0,0,1)\), and where \[ (x_1,x_2,x_3)^2=(-x_2x_3,x_3x_1,-k^2x_1x_2), \] which defines a commutative product over \({\mathbb R}^3\) [see \textit{S. Walcher}, Algebras and differential equations, Hadronic Press, Palm Harbor, FL (1991; Zbl 0791.17002 resp. Zbl 0754.17002)]. The purpose of the paper under review is to take advantage of the commutative algebra thus defined over \({\mathbb R}^3\), denoted by \(J(k)\), to give a recursive formula for the power series expansion of the Jacobi elliptic functions. To achieve this, it is first proved that \(J(k)\) is a simple algebra equipped with a distinguished \({\mathbb Z}_2\)-grading, and that the coefficients of the power series expansion belong alternatively to the even and odd parts. A discussion over arbitrary fields of characteristic zero is also carried out.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 17A01
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1139783
zbMATH Keywords: Jacobi elliptic functions
zbMATH Keywords: quadratic differential equations
zbMATH Keywords: graded algebra
MaRDI profile type: Publication
full work available at URL: jabr.1997.7253
OpenAlex ID: W1971548221
cites work: Q3285764
cites work: Q4147174
cites work: Quadratic dynamical systems and algebras
cites work: Jacobi elliptic algebras of SO(3)
cites work: Q3682641
cites work: Lie groups and algebras with applications to physics, geometry, and mechanics
cites work: Q4002714
cites work: Q3844537
DOI: 10.1006/JABR.1997.7253
description: scientific article; zbMATH DE number 1139783