Multipole matrix elements of Green function of Laplace equation

摘要

Multipole matrix elements of Green function of Laplace equation arecalculated. The multipole matrix elements of Green function in electrostaticsdescribe potential on a sphere which is produced by a charge distributed on thesurface of a different (possibly overlapping) sphere of the same radius. Thematrix elements are defined by double convolution of two spherical harmonicswith the Green function of Laplace equation. The method we use relies on thefact that in the Fourier space the double convolution has simple form.Therefore we calculate the multipole matrix from its Fourier transform. Animportant part of our considerations is simplification of the three dimensionalFourier transformation of general multipole matrix by its rotational symmetryto the one-dimensional Hankel transformation.