An Operational Representation of the Addition Theorems for Spherical Waves

In problems of wave propagation, it is sometimes necessary to transform the wave-functions in one coordinate system into another system which is convenient for the boundary-value problem in question. Such a transformation is usually achieved by an "addition theorem" which relates the eigen functions of the two systems. Sato has obtained an addition theorem for h_n(1)(kR)P^m_n(cos θ)e^(imφ) valid for a translation of the origin along the Z-axis. Friedman and Russek obtained addition theorems for standing, converging and diverging spherical waves under conditions of a combined rotation and translation of the spherical coordinate system. It will be shown here that their results can be condensed into a relatively simple form which is advantageous in many applications. For the reader's benefit, we have preserved the notation of Friedman and Russek with minor changes.