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Spherical Harmonic Closure Relations

The sum of the absolute squares of the Spherical Harmonics $Y_l^m(\theta, \phi)$ over all values of $m$ is

\begin{displaymath}
\sum_{m=-l}^l \vert Y_l^m(\theta,\phi)\vert^2 = {2l+1\over 4\pi}.
\end{displaymath}

The double sum over $m$ and $l$ is given by

\begin{eqnarray*}
\sum_{l=0}^\infty \sum_{m=-l}^l Y_l^m(\theta_1,\phi_1){Y_l^m}^...
...\
&=& \delta(\cos\theta_1-\cos \theta_2)\delta(\phi_1-\phi_2),
\end{eqnarray*}



where $\delta(x)$ is the Delta Function.




© 1996-9 Eric W. Weisstein
1999-05-26