Mallows' integral and some generalizations
type: Item
instance of: scholarly article
title: Mallows' integral and some generalizations
zbMATH Open document ID: 0674.33001
DOI: 10.1016/0022-247X(89)90127-3
author: S. H. Smith
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: Almost ten years ago Mallows proposed the problem of directly evaluating the integral \[ \int^{\pi}_{0}[\sin u(\sin \alpha u)^{-\alpha}(\sin \beta u)^{-\beta}]^ t du, \] when \(\alpha +\beta =1\). Its value is \(\Gamma (t+1)/\Gamma (\alpha t+1)\Gamma (\beta t+1)\), so it is clearly a type of beta integral. Mallows' method of deriving this was to solve a probability problem two ways. A second evaluation using Lagrange inversity was found by Evans, Ismail and Stanton. The present paper derives this and extensions using a change of variables in Cauchy's beta integral. Related integrals are evaluated using this method and some contour bending.
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 4103457
zbMATH Keywords: Cauchy theorem
zbMATH Keywords: beta integral
reviewed by: R. Askey
MaRDI profile type: Publication
cites work: Coefficients in Expansions of Certain Rational Functions
full work available at URL: 0022-247x(89)90127-3
OpenAlex ID: W2076386633
description: scientific article; zbMATH DE number 4103457