Asymptotic expansion of a quadruple integral involving a Bessel function
type: Item
instance of: scholarly article
title: Asymptotic expansion of a quadruple integral involving a Bessel function
zbMATH Open document ID: 0723.41029
DOI: 10.1016/0377-0427(90)90369-B
author: J. P. McClure
author: Roderick Wong
published in: Journal of Computational and Applied Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The quadruple integral \[ Q(\lambda):=\int^{\pi /2}_{0}...\int^{\pi /2}_{0}I_ 0(\lambda [\cos \alpha_ 1\quad \cos \alpha_ 2+\cos \alpha_ 3\quad \cos \alpha_ 4])d\alpha_ 1d\alpha_ 2d\alpha_ 3d\alpha_ 4, \] which occurs in crystallographic problems, is investigated with respect to its asymptotic behavior for \(\lambda\to \infty\), \(\lambda\in {\mathbb{R}}\). Here \(I_ 0\) denotes the ordinary Bessel function of order zero. The authors develop a new method of constructing an asymptotic expansion, where the first step reduces the integral to a finite Hankel transform. The second step consists of deriving an asymptotic expansion for this finite transform. The details of this approach are quite complicated. A lot of special results from classical analysis, e.g. analytic properties of some Mellin transforms and of the Gamma function, analytic continuation techniques for some meromorphic functions, are necessary for proving the asymptotic results. Some more interesting examples are included in the paper in order to show the wide applicability of the new method.
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 4192347
zbMATH Keywords: crystallographic problems
zbMATH Keywords: Bessel function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4192347