## Abstract and Applied Analysis

### The Analysis of Contour Integrals

#### Abstract

For any $n$, the contour integral $y={\text{cosh}}^{n+1}x{\oint{}}_{C}\text{(cosh}(zs)/{(\text{sinh}\text{\,}z-\text{sinh}\text{\,}x)}^{n+1})d{z,}$ ${s}^{2}=-{\lambda{}}$, is associated with differential equation ${d}^{2}y(x)/d{x}^{2}+({\lambda{}}+n(n+1)/{\text{cosh}}^{2}x)y(x)=0$. 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 765920, 12 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969152

Digital Object Identifier
doi:10.1155/2008/765920

Mathematical Reviews number (MathSciNet)
MR2393117

Zentralblatt MATH identifier
1182.34018

#### Citation

Tanriverdi, Tanfer; Mcleod, JohnBryce. The Analysis of Contour Integrals. Abstr. Appl. Anal. 2008 (2008), Article ID 765920, 12 pages. doi:10.1155/2008/765920. https://projecteuclid.org/euclid.aaa/1220969152

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