The analysis of contour integrals (Q938359)

The authors show that, for any positive integer \(n\), the contour integrals \[ y_1(x)=\cosh^{n+1}x\oint_C\frac{\cosh(zs)}{(\sinh z-\sinh x)^{n+1}}dz,\quad s^2=-\lambda, \] where the contour \(C\) is taken round the point \(z = x\) and no other zero of \(\sinh z-\sinh x\) and \[ y_2(x)=\cosh^{n+1}x\oint_C\frac{\sinh(zs)}{(\sinh z-\sinh x)^{n+1}}dz,\quad s^2=-\lambda, \] satisfy the differential equation \[ \frac{d^2y}{dx^2}+(\lambda+n(n+1)\text{sech}^2x)y(x)=0.\tag{\(*\)} \] For \(n = 1\), explicit solutions of (\(*\)) are obtained. Further, for \(n = 1\), the results concerning eigenvalues, eigenfunctions, spectral function and eigenfunction expansions are discussed. In particular, following results are obtained: Eigenvalues with associated boundary conditions \(y(0)=y(b)=0\) are the zeros of \(\sqrt{\lambda}\tan(b\sqrt{\lambda})+\tanh(b)=0\). Furthermore, one and only one eigenvalue lies in the interval \[ (k-\frac12)\pi<b\sqrt{\lambda}<(k+\frac12)\pi\tag{\(**\)} \] for every integer \(k\neq0\). The eigenvalues with \(y'(0)=y'(b)=0\) are the zeros of \(\lambda\tan(b\sqrt{\lambda})+\text{sech}^2(b)\tan(b\sqrt{\lambda})+\sqrt{\lambda}\tanh(b)=0\). Moreover, there exists one and only one eigenvalue lying in the interval (\(**\)) for every integer \(k\neq0\).