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arXiv · 0910.1492

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

Radoslaw Szmytkowski·Sebastian Bielski
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Abstract

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

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Radoslaw Szmytkowski, Sebastian Bielski. 2009-10-13. An orthogonality relation for the Whittaker functions of the second kind of imaginary order. https://arxiv.org/abs/0910.1492

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