Difference between revisions of "Spherical harmonics"

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m (Another slight tidy.)
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The '''spherical harmonics''' <math>Y_l^m (\theta,\phi)</math> are the angular portion of the solution to [[Laplace's equation]] in spherical coordinates.
 
The '''spherical harmonics''' <math>Y_l^m (\theta,\phi)</math> are the angular portion of the solution to [[Laplace's equation]] in spherical coordinates.
 +
They are given by
 +
:<math>Y_l^m  (\theta,\phi) =
 +
(-1)^m \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}
 +
P^m_n(\cos\theta) e^{i m \phi},</math>
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where <math> P^m_n </math> is the [[associated Legendre function]].
 +
 
The first few spherical harmonics are given by:
 
The first few spherical harmonics are given by:
  

Latest revision as of 12:54, 20 June 2008

The spherical harmonics Y_l^m (\theta,\phi) are the angular portion of the solution to Laplace's equation in spherical coordinates. They are given by

Y_l^m  (\theta,\phi) =
(-1)^m \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}}
P^m_n(\cos\theta) e^{i m \phi},

where  P^m_n is the associated Legendre function.

The first few spherical harmonics are given by:

Y_0^0 (\theta,\phi) = \frac{1}{2} \frac{1}{\sqrt{\pi}}
Y_1^{-1} (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{-i\phi}
Y_1^0 (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos \theta
Y_1^1 (\theta,\phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{i\phi}

See also[edit]

References[edit]