An orthogonality relation for the Whittaker functions of the second kind of imaginary order

Abstract

An orthogonality relation for the Whittaker functions of the second kind of imaginary order, $W_{\kappa,\mathrm{i}\mu}(x)$, with $\mu\in\mathbb{R}$, is investigated. The integral $\int_{0}^{\infty}\mathrm{d}x\: x^{-2}W_{\kappa,\mathrm{i}\mu}(x)W_{\kappa,\mathrm{i}\mu'}(x)$ is shown to be proportional to the sum $\delta(\mu-\mu')+\delta(\mu+\mu')$, where $\delta(\mu\pm\mu')$ is the Dirac delta distribution. The proportionality factor is found to be $\pi^{2}/[\mu\sinh(2\pi\mu)\Gamma({1/2}-\kappa+\mathrm{i}\mu) \Gamma({1/2}-\kappa-\mathrm{i}\mu)]$. For $\kappa=0$ the derived formula reduces to the orthogonality relation for the Macdonald functions of imaginary order, discussed recently in the literature.