The analysis of contour integrals
type: Item
instance of: scholarly article
title: The analysis of contour integrals
zbMATH Open document ID: 1182.34018
DOI: 10.1155/2008/765920
author: J. B. McLeod
author: Tanfer Tanriverdi
published in: Abstract and Applied Analysis
publication date: 2008-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/45267
review text: The authors show that, for any positive integer \(n\), the contour integrals \[ y_1(x)=\cosh^{n+1}x\oint_C\frac{\cosh(zs)}{(\sinh z-\sinh x)^{n+1}}dz,\quad s^2=-\lambda, \] where the contour \(C\) is taken round the point \(z = x\) and no other zero of \(\sinh z-\sinh x\) and \[ y_2(x)=\cosh^{n+1}x\oint_C\frac{\sinh(zs)}{(\sinh z-\sinh x)^{n+1}}dz,\quad s^2=-\lambda, \] satisfy the differential equation \[ \frac{d^2y}{dx^2}+(\lambda+n(n+1)\text{sech}^2x)y(x)=0.\tag{\(*\)} \] For \(n = 1\), explicit solutions of (\(*\)) are obtained. Further, for \(n = 1\), the results concerning eigenvalues, eigenfunctions, spectral function and eigenfunction expansions are discussed. In particular, following results are obtained: Eigenvalues with associated boundary conditions \(y(0)=y(b)=0\) are the zeros of \(\sqrt{\lambda}\tan(b\sqrt{\lambda})+\tanh(b)=0\). Furthermore, one and only one eigenvalue lies in the interval \[ (k-\frac12)\pi<b\sqrt{\lambda}<(k+\frac12)\pi\tag{\(**\)} \] for every integer \(k\neq0\). The eigenvalues with \(y'(0)=y'(b)=0\) are the zeros of \(\lambda\tan(b\sqrt{\lambda})+\text{sech}^2(b)\tan(b\sqrt{\lambda})+\sqrt{\lambda}\tanh(b)=0\). Moreover, there exists one and only one eigenvalue lying in the interval (\(**\)) for every integer \(k\neq0\).
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 5313170
zbMATH Keywords: contour integral
zbMATH Keywords: second order ordinary differential equation
zbMATH Keywords: eigenvalues
zbMATH Keywords: eigenfunctions
zbMATH Keywords: spectral function
zbMATH Keywords: eigenfunction expansion
Wikidata QID: Q58644392
MaRDI profile type: Publication
OpenAlex ID: W2052865415
cites work: Q3279399
cites work: The approach of solutions of nonlinear diffusion equations to travelling front solutions
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cites work: Wave solutions for a discrete reaction-diffusion equation
cites work: Generalization of the eigenvalues by contour integrals
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description: scientific article; zbMATH DE number 5313170