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    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 Preview

    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 ...

    Solution Summary

    This solution shows how Green's Function and the Spherical Harmonics Expansion are equivalent.