Laplace Series

A function $f(\theta,\phi)$ expressed as a double sum of Spherical Harmonics is called a Laplace series. Taking $f$ as a Complex Function,

$$f(\theta, \phi) = \sum_{l=0}^\infty \sum_{m=-1}^l a_{lm}Y_l^m(\theta,\phi).$$ (1)

Now multiply both sides by ${Y_{l'}^{m'}}^*\sin\theta$ and integrate over $d\theta$ and $d\phi$.

$\int^{2\pi}_0 \int^{\pi}_0 f(\theta,\phi){Y_{l'}^{m'}}^*\sin \theta\,d\theta\,d\phi$
$= \sum_{l=0}^\infty \sum_{m=-l}^l \! a_{lm}\int^{2\pi}_0\! \int^{\pi}_0 {Y_{l'}^{m'}}^*(\theta,\phi) Y_l^m(\theta,\phi)\sin \theta\,d\theta\,d\phi.$
	(2)

Now use the Orthogonality of the Spherical Harmonics

$$\int^{2\pi}_0\!\int^{\pi}_0 Y_l^m(\theta,\phi){Y_{l'}^{m'}}^*\sin\theta\,d\theta\,d\phi=\delta_{mm'}\delta_{ll'},$$ (3)

so (2) becomes

$$\int^{2\pi}_0 \int^{\pi}_0 f(\theta,\phi){Y_{l'}^{m'}}^*\sin... ...}^\infty \sum_{m=-1}^l a_{lm}\delta_{mm'}\delta_{ll'}= a_{lm},$$ (4)

where $\delta_{mn}$ is the Kronecker Delta.

For a Real series, consider

$$f(\theta,\phi) = \sum_{l=0}^\infty\sum_{m=-1}^l [C_l^m\cos(m\phi)+S_l^m\sin(m\phi)] P_l^m(\cos\theta).$$ (5)

Proceed as before, using the orthogonality relationships

$\int^{2\pi}_0 \int^{\pi}_0 P_l^m(\cos\theta)\cos(m\phi)P_{l'}^{m'}(\cos\theta)\cos(m'\phi)\sin(\theta)\,d\theta\,d\phi$
$= - {2\pi (l+m)!\over (2l+1)(l-m)!}\delta_{mm'}\delta_{ll'}\quad$	(6)

$\int^{2\pi}_0 \int^{\pi}_0 P_l^m(\cos\theta)\sin(m\phi)P_{l'}^{m'}(\cos\theta)\sin(m'\phi)\sin\theta\,d\theta \,d\phi$
$= - {2\pi(l+m)!\over (2l+1)(l-m)!}\delta_{mm'}\delta_{ll'}.\quad$	(7)

So $C_l^m$ and $S_l^m$ are given by

$$C_l^m = - {(2l+1)(l-m)!\over 2\pi (l+m)!}\int^{2\pi}_0 \int^{\pi}_0 f(\theta, \phi)P_l^m\cos\theta\cos(m\phi)\sin\theta\,d\theta\,d\phi$$ (8)

$$S_l^m = - {(2l+1)(l-m)!\over 2\pi (l+m)!} \int^{2\pi}_0 \int^{\pi}_0 f(\theta, \phi)P_l^m\cos\theta\sin(m\phi)\sin\theta\,d\theta\,d\phi.$$ (9)