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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,

\begin{displaymath}
f(\theta, \phi) = \sum_{l=0}^\infty \sum_{m=-1}^l a_{lm}Y_l^m(\theta,\phi).
\end{displaymath} (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
\begin{displaymath}
\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'},
\end{displaymath} (3)

so (2) becomes


\begin{displaymath}
\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},
\end{displaymath} (4)

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


For a Real series, consider


\begin{displaymath}
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).
\end{displaymath} (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


$\displaystyle C_l^m$ $\textstyle =$ $\displaystyle - {(2l+1)(l-m)!\over 2\pi (l+m)!}\int^{2\pi}_0 \int^{\pi}_0 f(\theta, \phi)P_l^m\cos\theta\cos(m\phi)\sin\theta\,d\theta\,d\phi$ (8)
$\displaystyle S_l^m$ $\textstyle =$ $\displaystyle - {(2l+1)(l-m)!\over 2\pi (l+m)!} \int^{2\pi}_0 \int^{\pi}_0 f(\theta, \phi)P_l^m\cos\theta\sin(m\phi)\sin\theta\,d\theta\,d\phi.$ (9)




© 1996-9 Eric W. Weisstein
1999-05-26