Comment on the orthogonality of the Macdonald functions of imaginary order
type: Item
instance of: scholarly article
title: Comment on the orthogonality of the Macdonald functions of imaginary order
zbMATH Open document ID: 1186.33009
author: Radosław Szmytkowski
author: Sebastian Bielski
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-02-19T00:00:00Z
review text: 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.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 5673268
zbMATH Keywords: Macdonald functions
zbMATH Keywords: orthogonality relations
zbMATH Keywords: Dirac delta distribution
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.10.035
OpenAlex ID: W2049721604
cites work: Q5443785
cites work: On the orthogonality of the MacDonald's functions
cites work: Q4382680
cites work: An orthogonality relation for the Whittaker functions of the second kind of imaginary order
cites work: Q5518575
cites work: Q3280242
DOI: 10.1016/J.JMAA.2009.10.035
description: scientific article; zbMATH DE number 5673268