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# Surface Harmonics and Solid Harmonics

A surface harmonic of degree (where is a non-negative integer), denoted , is defined as a well-behaved solution to

 (12.36)

on the surface of a sphere (i.e., constant). Here, , , are standard spherical coordinates, and is the Laplacian operator. It follows that (Love 1927)

 (12.37)

where the and are arbitrary coefficients, and the associated Legendre functions.

A solid harmonic of degree (where is a non-negative integer), denoted , is defined as a well-behaved solution to

 (12.38)

in the interior of a sphere (i.e., the region constant). It follows that (Love 1927)

 (12.39)

In particular, the functions and , introduced in Section 12.3, are solid harmonics of degree . Note that the Cartesian coordinates (where runs from to ) are solid harmonics of degree . Moreover, is a solid harmonic of degree . Finally, , specified in Equation (12.35), is a solid harmonic of degree 2.

The following results regarding solid harmonics are useful (Love 1927):

 (12.40) (12.41) (12.42)

Here, use has been made of the Einstein summation convention (Riley 1974).

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Richard Fitzpatrick 2016-03-31