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

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

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.