A function
expressed as a double sum of Spherical Harmonics is called a
Laplace series. Taking as a Complex Function,
|
(1) |
Now multiply both sides by
and integrate over and .
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|
|
|
|
(2) |
Now use the Orthogonality of the Spherical Harmonics
|
(3) |
so (2) becomes
|
(4) |
where is the Kronecker Delta.
For a Real series, consider
|
(5) |
Proceed as before, using the orthogonality relationships
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|
|
(6) |
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|
|
(7) |
So and are given by
© 1996-9 Eric W. Weisstein
1999-05-26