In spherical coordinates, the scale factors are , , , and the separation functions are , , , giving a Stäckel determinant of .
The Laplacian is
(1)
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To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing
(2)
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Then the Helmholtz differential equation becomes
(3)
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Now divide by ,
(4)
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(5)
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The solution to the second part of (5) must be sinusoidal, so the differential equation is
(6)
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which has solutions which may be defined either as a complex function with , ...,
(7)
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or as a sum of real sine and cosine functions with , ...,
(8)
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(9)
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The radial part must be equal to a constant
(10)
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(11)
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But this is the Euler differential equation, so we try a series solution of the form
(12)
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Then
(13)
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(14)
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(15)
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This must hold true for all powers of . For the term (with ),
(16)
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which is true only if and all other terms vanish. So for , . Therefore, the solution of the component is given by
(17)
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Plugging (17) back into (◇),
(18)
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(19)
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which is the associated Legendre differential equation for and , ..., . The general complex solution is therefore
(20)
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where
(21)
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are the (complex) spherical harmonics. The general real solution is
(22)
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Some of the normalization constants of can be absorbed by and , so this equation may appear in the form
(23)
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where
(24)
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(25)
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are the even and odd (real) spherical harmonics. If azimuthal symmetry is present, then is constant and the solution of the component is a Legendre polynomial . The general solution is then
(26)
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