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 ${\int }_0^{\infty }\mathrm{d}x{x}^{-2}{W}_{\kappa ,\mathrm{i}\mu }(x)W_{\kappa ,\mathrm{i}{\mu }^{\prime }}(x)$ is shown to be proportional to the sum $\delta (\mu -{\mu }^{\prime })+\delta (\mu +{\mu }^{\prime })$, where $\delta (\mu \pm {\mu }^{\prime })$ 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.

LaTeX, 5 pages, content revised

References in corpus (1)

• Comment on the orthogonality of the Macdonald functions of imaginary order

Cited by in corpus (1)

• Comment on the orthogonality of the Macdonald functions of imaginary order