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.