On the evaluation of the integral over the product of two spherical Bessel functions

Abstract

The integral Il,l′(k,k′)=∫∞0jl (kr)jl′(k′r)r 2 dr, in which the spherical Bessel functions jl(kr) are the radial eigenfunctions of the three-dimensional wave equation in spherical coordinates, is evaluated in terms of distributions, in particular, step functions and delta functions. It will be shown that the behavior of Il,l′ is very different in the cases l−l′ even (0, ±2, ±4, ...) and l−l′ odd (±1, ±3, ...). For l−l′ even it is expressed in terms of the delta function, step functions, and Legendre polynomials. For l−l′ odd it is expressed in terms of Legendre functions of the second kind and step functions; no delta functions appear.