Q919154
type: Item
instance of: scholarly article
title: Eine neue Funktionalgleichung zur Bestimmung elliptischer Integrale erster Gattung und ihrer Umkehrungen. (A new functional equation for the definition of elliptic integrals of the first kind and their inverses)
zbMATH Open document ID: 0706.33016
published in: Elemente der Mathematik
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/141405
review text: In order to calculate \[ F(x;m)=\int^{x}_{0}(1-t^ 2)^{-1/2}(1- mt^ 2)^{-1/2} dt \] in the left neighbourhood of 1, the author offers the functional equation \[ F(x;m)=K(m)-F((1-x^ 2)^{1/2}(1-mx^ 2)^{-1/2};m), \] where \[ K(m)=(\pi /2)\sum^{\infty}_{n=0}((2n)!)^ 2 2^{-4n}(n!)^{-4} m^ n. \] For negative m, the equation \[ F(x;m)=(1-m)^{-1/2}F((1-m)^{1/2}(1- mx^ 2)^{-1/2}x;m(m-1)^{-1}) \] is stated and used. A functional equation for the inverse function of F with respect to the first variable is also offered. Two numerical examples \((m=\) and \(m=-2)\) are calculated in detail.
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 4159101
zbMATH Keywords: generating functions
zbMATH Keywords: areas
zbMATH Keywords: limits
zbMATH Keywords: inverse function
author: Wolfgang Dreyer
MaRDI profile type: Publication
label: Eine neue Funktionalgleichung zur Bestimmung elliptischer Integrale erster Gattung und ihrer Umkehrungen. (A new functional equation for the definition of elliptic integrals of the first kind and their inverses)
description: scientific article; zbMATH DE number 4159101