A numerical evaluation of very oscillatory infinite integrals by asymptotics
type: Item
instance of: scholarly article
title: A numerical evaluation of very oscillatory infinite integrals by asymptotics
zbMATH Open document ID: 0714.65022
DOI: 10.1016/0096-3003(90)90126-N
author: Bui Doan Khanh
published in: Applied Mathematics and Computation
publication date: 1990-01-01T00:00:00Z
review text: Let f be a real smooth function integrable at infinity and satisfying a linear differential equation \(f(t)=p(t)f'(t)\). Assume that p has an asymptotic expansion p(t)\(\sim \sum^{r}_{-\infty}a_ kt^ k\) with \(r\leq 0\) and \(a_ r\neq 0\) and all derivatives of p have asymptotic expansions obtained by differentiating the expansion of p term by term. If p(t)f(t)\(\to 0\) as \(t\to \infty\) one obtains by integrating by parts n times a representation \(\int^{\infty}_{x}f(t)dt=(A_ 1(x)+...+A_ n(x))f(x)+R(x)\) where the \(A_ n\) are functions of p and its derivatives. The remainder R(x) is claimed to be \(o(x^{-n-r})\) in theorem 1 but the proof yields only \(o(x^{1-n-r})\). With some modifications this can also be applied to the case \(r=1\) and to the case where f satisfies a linear differential equation of the second order. This is applied to the evaluation of integrals of functions of the type \(f(t)=h(t)\exp (b(t)+ia(t))\) where a, b, h have suitable asymptotic expansions.
reviewed by: Rudolf Wegmann
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 4176330
zbMATH Keywords: very oscillatory infinite integrals
zbMATH Keywords: asymptotic expansion
MaRDI profile type: Publication
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cites work: Q3026211
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full work available at URL: 0096-3003(90)90126-n
OpenAlex ID: W1976496733
Wikidata QID: Q127847512
description: scientific article; zbMATH DE number 4176330