On the Integral Identities Consisting of Two Spherical Bessel Functions

Abstract

When deriving dyadic Green's functions for the spherical structures with gyrotropic or bianisotropic materials, an integral whose integrand function consists of two spherical Bessel functions and a power function needs to be evaluated. Therefore, this paper revisits thoroughly the evaluation of the integral of Il,l'(kappa,kappa'). Starting from pointing out an error, it provides the correct solution to the integral in spherical coordinates in terms of distribution, in particular, step functions and delta functions. The formulation is further extended to a more generalized integral Hl,l'lambda(kappa,kappa'); and it is newly found that the solution to the generalized integral varies differently in the cases of even and odd values of l-l'. The mistakes that we found in the previous literature can also be proved easily by some of our intermediate solutions