Spherical Harmonics: Difference between revisions

Spherical Harmonics (view source)

622 bytes added , 6 November 2019

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 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>
+* <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
+: and <math>K_l^m = \sqrt{\frac{(2l+1)(l-|m|)!}{4 \pi (l+|m|)!}}</math>
+For a real valued basis,
+* <math>y_l^m = \begin{cases}
+\sqrt{2}\operatorname{Re}(Y_l^m) & m > 0\\
+\sqrt{2}\operatorname{Im}(Y_l^m) & m < 0\\
+Y_l^0 & m = 0
+\end{cases}
+= \begin{cases}
+\sqrt{2} K_l^m \cos(m \varphi) P_l^m(\cos\theta) & m > 0\\
+\sqrt{2} K_l^m \sin(|m| \varphi) P_l^{|m|}(\cos\theta) & m < 0\\
+K_l^0 P_l^0 (\cos \theta) & m = 0\\
+\end{cases}</math>
 ==Applications==