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Spherical Harmonics: Difference between revisions
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==Definition== | ==Definition== | ||
Spherical Harmonics are a set of orthonormal basis functions defined on the sphere.<br> | Spherical Harmonics are a set of orthonormal basis functions defined on the sphere.<br> | ||
Below are some explicit formulas for Laplace spherical harmonics stolen from <ref name="stupidsh">Peter-Pike Sloan, [http://www.ppsloan.org/publications/StupidSH36.pdf Stupid Spherical Harmonics (SH) Tricks]</ref> | Below are some explicit formulas for Laplace spherical harmonics stolen from <ref name="stupidsh">Peter-Pike Sloan, [http://www.ppsloan.org/publications/StupidSH36.pdf Stupid Spherical Harmonics (SH) Tricks]</ref><br> | ||
There are <math>2l+1</math> functions for each band.<br> | |||
* <math>Y_l^m(\theta, \varphi) = K_l^m e^{i m \varphi} P_l^{|m|} \cos(\theta)</math> for <math>-l \leq m \leq l</math> | * <math>Y_l^m(\theta, \varphi) = K_l^m e^{i m \varphi} P_l^{|m|} \cos(\theta)</math> for <math>-l \leq m \leq l</math> | ||
: where <math>P_l^m</math> are the associated Legendre Polynomials | : where <math>P_l^m</math> are the associated Legendre Polynomials | ||
: and <math>K_l^m = \sqrt{\frac{(2l+1)(l-|m|)!}{4 \pi (l+|m|)!}}</math> | : and <math>K_l^m = \sqrt{\frac{(2l+1)(l-|m|)!}{4 \pi (l+|m|)!}}</math> | ||
: l is the band, m is the function | |||
For a real valued basis, | For a real valued basis, | ||