Green's Function and Spherical Harmonics
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Please see the attached file. It is taken from "Classical Electrodynamic 3rd by Jackson".
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Solution Summary
This solution shows how Green's Function and the Spherical Harmonics Expansion are equivalent.
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We know that the Green's function for a sphere with Dirichlet's boundary conditiosn is
(1.1)
If the angle between and r is then it can be written as:
(1.2)
On the surface of the sphere:
(1.3)
The potential is given by:
(1.4)
Since there is no excess free charge we have
Since this is Green's function with ...
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