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Spherical Harmonics
The spherical harmonics,
, are the angular portions of the global
solutions to Laplace's equation in standard spherical coordinates,
,
,
. Here,
is a nonnegative
integer (known as the degree), and
is an integer (known as the order) lying in the range
. The spherical
harmonics are well behaved and single valued functions that satisfy the
differential equation

(307) 
and take the form^{}

(308) 
It follows from Equation (295) that

(309) 
The
satisfy the orthonormality constraint

(310) 
where
is a an element of solid angle, and the integral is taken
over all solid angle.
Note that the spherical harmonics form a complete set of angular functions.
All of the spherical harmonics of degree less than 3 are listed below:
Consider two spherical coordinate systems,
,
,
and
,
,
, whose origins coincide, but
whose polar axes subtend an angle
with respect to one another. It follows that

(320) 
Moreover, the socalled addition theorem for spherical harmonics states that^{}

(321) 
Next: Laplace's Equation in Spherical
Up: Potential Theory
Previous: Associated Legendre Functions
Richard Fitzpatrick
20140627