Addition Theorems for Spherical Bessel Functions and Vector Harmonics in Spherical Coordinates

Abstract

By means of the theory of representation of 3-dimensional Euclidean motions, generalized spherical Bessel functions are defined y matrix elements, and the addition theorem as well as the generalized Fourier-Bessel integral theorem for the functions is established. Using the concept of l -vector, the spherical and the solid l -vector harmonics are defined and their properties are studied. The addition theorem is represented as a two-point-tensor formula in terms of l -vector harmonics. The special case l =1 gives a unified description of the ordinary vector harmonic functions which have been defined diversely. Solid vector harmonics satisfy the vector Helmholtz equation, and the addition formula for a generalized spherical Hankel function which is an isotropic tensor field gives the transverse or the longitudinal part of the Green's dyadic for free space.