Jacobian elliptic functions as inverses of an integral
type: Item
instance of: scholarly article
title: Jacobian elliptic functions as inverses of an integral
zbMATH Open document ID: 1098.33013
author: B. C. Carlson
published in: Journal of Computational and Applied Mathematics
publication date: 2005-02-23T00:00:00Z
review text: For \(a_j, b_j \in \mathbb{R}, j=1,2,\) let \(w(t)=(a_1+b_1 t^2)(a_2+b_2 t^2).\) The author studies Jacobian elliptic functions as special cases of the inverses of the elliptic integrals \(\int_x^y w(t)^{-1/2} dt\) in which either \(y=0\) or \(x=\infty\) or \(a_1+b_1 y^2=0\) or \(a_2+b_2 x^2=0.\) The author gives a new unified treatment and shows, in particular, that in each of these four cases the other limit of the integration is determined as the inverse function of the integral by the products \(a_1 b_2\) and \(a_2 b_1 .\) The proof relies on the theory of symmetric elliptic integrals due to the author [Special functions of applied mathematics. (New York etc. Academic Press (1977; Zbl 0394.33001)].
reviewed by: M. Vuorinen
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 2137434
zbMATH Keywords: elliptic functions
MaRDI profile type: Publication
full work available at URL: j.cam.2004.05.001
OpenAlex ID: W4206643159
cites work: Q4175520
DOI: 10.1016/J.CAM.2004.05.001
Recommended article: Inequalities involving Jacobian elliptic functions and their inverses
Recommended article: Expression of elliptic functions in terms of Jacobi functions
Recommended article: Analogues of Jacobi's inversion formula for the incomplete elliptic integral of the first kind
Recommended article: A functional equation and Jacobian elliptic functions
Recommended article: Q3819466
Recommended article: Q5705769
Recommended article: Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions
Recommended article: An inverse convergence approach for arguments of Jacobian elliptic functions
Recommended article: Inversion formulas for the \(j\)-function around elliptic points
description: scientific article; zbMATH DE number 2137434