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: | , |
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2008/765920 |
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| _version_ | 1832567967697600512 |
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| author | Tanfer Tanriverdi JohnBryce Mcleod |
| author_facet | Tanfer Tanriverdi JohnBryce Mcleod |
| author_sort | Tanfer Tanriverdi |
| collection | DOAJ |
| 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. |
| format | Article |
| id | doaj-art-d4a94f774d804deb9ee4d5d0dd9e1981 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-d4a94f774d804deb9ee4d5d0dd9e19812025-02-03T01:00:02ZengWileyAbstract 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 |
| spellingShingle | Tanfer Tanriverdi JohnBryce Mcleod The Analysis of Contour Integrals Abstract and Applied Analysis |
| title | 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_short | The Analysis of Contour Integrals |
| title_sort | analysis of contour integrals |
| url | http://dx.doi.org/10.1155/2008/765920 |
| work_keys_str_mv | AT tanfertanriverdi theanalysisofcontourintegrals AT johnbrycemcleod theanalysisofcontourintegrals AT tanfertanriverdi analysisofcontourintegrals AT johnbrycemcleod analysisofcontourintegrals |