The Analysis of Contour Integrals

The Analysis of Contour Integrals

For any 𝑛, the contour integral 𝑦=cosh𝑛+1𝑥∮𝐶(cosh(𝑧𝑠)/(sinh𝑧−sinh𝑥)𝑛+1𝑑𝑧,𝑠2=−𝜆, is associated with differential equation 𝑑2𝑦(𝑥)/𝑑𝑥2+(𝜆+𝑛(𝑛+1)/cosh2𝑥)𝑦(𝑥)=0. Explicit solutions for 𝑛=1 are obtained. For 𝑛=1, eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored....

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Main Authors: Tanfer Tanriverdi, JohnBryce Mcleod
Format: Article
Language:English
Published: Hindawi Limited 2008-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2008/765920
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spelling doaj-d4a94f774d804deb9ee4d5d0dd9e19812020-11-24T22:39:51ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092008-01-01200810.1155/2008/765920765920The Analysis of Contour IntegralsTanfer Tanriverdi0JohnBryce Mcleod1Department of Mathematics, Harran University, Osmanbey Campus, Sanlurfa 63100, TurkeyDepartment of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USAFor any 𝑛, the contour integral 𝑦=cosh𝑛+1𝑥∮𝐶(cosh(𝑧𝑠)/(sinh𝑧−sinh𝑥)𝑛+1𝑑𝑧,𝑠2=−𝜆, is associated with differential equation 𝑑2𝑦(𝑥)/𝑑𝑥2+(𝜆+𝑛(𝑛+1)/cosh2𝑥)𝑦(𝑥)=0. Explicit solutions for 𝑛=1 are obtained. For 𝑛=1, eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored. This differential equation which does have solution in terms of the trigonometric functions does not seem to have been explored and it is also one of the purposes of this paper to put it on record.http://dx.doi.org/10.1155/2008/765920
collection DOAJ
language English
format Article
sources DOAJ
author Tanfer Tanriverdi
JohnBryce Mcleod
spellingShingle Tanfer Tanriverdi
JohnBryce Mcleod
The Analysis of Contour Integrals
Abstract and Applied Analysis
author_facet Tanfer Tanriverdi
JohnBryce Mcleod
author_sort Tanfer Tanriverdi
title The Analysis of Contour Integrals
title_short The Analysis of Contour Integrals
title_full The Analysis of Contour Integrals
title_fullStr The Analysis of Contour Integrals
title_full_unstemmed The Analysis of Contour Integrals
title_sort analysis of contour integrals
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2008-01-01
description For any 𝑛, the contour integral 𝑦=cosh𝑛+1𝑥∮𝐶(cosh(𝑧𝑠)/(sinh𝑧−sinh𝑥)𝑛+1𝑑𝑧,𝑠2=−𝜆, is associated with differential equation 𝑑2𝑦(𝑥)/𝑑𝑥2+(𝜆+𝑛(𝑛+1)/cosh2𝑥)𝑦(𝑥)=0. Explicit solutions for 𝑛=1 are obtained. For 𝑛=1, eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored. This differential equation which does have solution in terms of the trigonometric functions does not seem to have been explored and it is also one of the purposes of this paper to put it on record.
url http://dx.doi.org/10.1155/2008/765920
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AT johnbrycemcleod theanalysisofcontourintegrals
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