Comment on the orthogonality of the Macdonald functions of imaginary order

TL;DR

This paper offers a simpler proof of the orthogonality relation for Macdonald functions of imaginary order by applying a technique from mathematical physics used for normalizing scattering wave functions.

Contribution

It introduces a more straightforward method to prove the orthogonality of Macdonald functions of imaginary order, simplifying previous proofs.

Findings

Orthogonality relation for Macdonald functions confirmed

Simpler proof technique demonstrated

Method applicable to other special functions

Abstract

Recently, Yakubovich [Opuscula Math. 26 (2006) 161--172] and Passian et al. [J. Math. Anal. Appl. doi:10.1016/j.jmaa.2009.06.067] have presented alternative proofs of an orthogonality relation obeyed by the Macdonald functions of imaginary order. In this note, we show that the validity of that relation may be also proved in a simpler way by applying a technique occasionally used in mathematical physics to normalize scattering wave functions to the Dirac delta distribution.