Please see the attached file. It is taken from "Classical Electrodynamic 3rd by Jackson".© BrainMass Inc. brainmass.com October 10, 2019, 7:46 am ad1c9bdddf
We know that the Green's function for a sphere with Dirichlet's boundary conditiosn is
If the angle between and r is then it can be written as:
On the surface of the sphere:
The potential is given by:
Since there is no excess free charge we have
Since this is Green's function with ...
This solution shows how Green's Function and the Spherical Harmonics Expansion are equivalent.