Abstract and Applied Analysis
Volume 2008 (2008), Article ID 765920, 12 pages
doi:10.1155/2008/765920
Research Article

The Analysis of Contour Integrals

Tanfer Tanriverdi1 and JohnBryce Mcleod2

1Department of Mathematics, Harran University, Osmanbey Campus, Sanlurfa 63100, Turkey
2Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

Received 10 November 2007; Accepted 19 January 2008

Academic Editor: Stephen L. Clark

Copyright © 2008 Tanfer Tanriverdi and JohnBryce Mcleod. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.