Numerical evaluation of a Cauchy principal value integral that arises in a problem involving the generation of instability waves
type: Item
instance of: scholarly article
title: Numerical evaluation of a Cauchy principal value integral that arises in a problem involving the generation of instability waves
zbMATH Open document ID: 0602.65009
DOI: 10.1016/0021-9991(85)90118-4
author: Avram Sidi
published in: Journal of Computational Physics
publication date: 1985-01-01T00:00:00Z
review text: The purpose of the present paper is to compute the principal value of an integral whose integrand is given in terms of Cauchy principal value integrals. For the integrand the author applies analytic methods as far as possible, using some special functions such as Dilogarithms. The main integral has the form \(\oint^{\infty}_{0}\Phi (t)/(t-1)dt\), with \(\Phi\) (1)\(\neq 0\). He first analyzes the behaviors of \(\Phi\) (t) at 0 and \(+\infty\), and then divides the interval \([0,+\infty)\) into three parts, say [0,1/2],[1/2,3/2], and \([3/2,+\infty)\). For the second interval involving principal value, he applies the Longman's idea \[ I=\oint^{a}_{-a}F(x)/x dx=\oint^{a}_{-a}[F(x)-F(-x)]/2x dx \] and uses Gauss-Legendre integration formula with even number of nodes to avoid the functional value 0/0 at the origin (corresponding to \(x=1)\). The result seems quite satisfactory.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 3972060
zbMATH Keywords: generation of instability waves
zbMATH Keywords: Cauchy principal value integrals
zbMATH Keywords: analytic methods
zbMATH Keywords: Dilogarithms
zbMATH Keywords: Gauss-Legendre integration formula
MaRDI profile type: Publication
full work available at URL: 0021-9991(85)90118-4
OpenAlex ID: W2056939914
cites work: Q3661894
cites work: Q3253375
cites work: The special functions and their approximations. Vol. I, II
cites work: The Dilogarithm Function of a Real Argument
cites work: On the Numerical Evaluation of Cauchy Principal Values of Integrals
cites work: Numerical evaluation of cauchy principal values of integrals
cites work: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals
description: scientific article; zbMATH DE number 3972060