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

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

2009-10-08

arxiv.org/abs/0910.1492v2

Mathematics

Classical Analysis and ODEs

LaTeX, 5 pages, content revised

Scientific paper

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.

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