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单词 Laplace Series
释义

Laplace Series

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 .

(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

(6)
(7)
So and are given by


(8)
(9)


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更新时间:2025/2/22 16:36:07