Comment on the orthogonality of the Macdonald functions of imaginary order
The orthogonality relation for MacDonald's functions \(K_{i\nu}(x)\) of imaginary order \[ \int_0^\infty \frac{K_{i\nu}(x)K_{i\nu'}(x)}{x}\,dx= \frac{\pi^2}{2\nu\sinh(\pi\nu)}\,\delta(\nu-\nu'), \quad \nu,\nu'>0) \tag{1} \] has been proven by \textit{S. B. Yakubovich} [Opusc. Math. 26, No.~1, 161--172 (2006; Zbl 1139.46035)] and also by \textit{A. Passian} et al. [ J. Math. Anal. Appl. 360, No.~2, 380--390 (2009; Zbl 1180.33006)]. In this paper the authors present still another proof of (1) which is simpler than the previous ones and based on a technique used in mathematical physics for normalization of scattering wave functions to the Dirac delta distribution.
scientific article; zbMATH DE number 5673268