# Laplace's equation in spherical co-ords

by benabean
Tags: coords, equation, laplace, spherical
 P: 32 I have a simple question about the general solution to Laplace's equation in spherical co-ords. The general solution is: $$u(r, \theta, \phi) = \sum^{\infty}_{l=0}\sum^{l}_{m=-l}\left(a_{lm}r^{l} + \frac{b_{lm}}{r^{l+1}}\right)P_{lm}(cos\theta)e^{im\phi}$$ (where the $$a_{lm}, b_{lm}$$ coefficients can be found using the boundary conditions in question.) My problem lies in trying to understand the limits on the summation $$\sum^{l}_{m=-l}$$. Can anyone offer any help on this please? Thanks for reading, b.
 P: 333 Probably due to the same reason why the irreducible representation of SO(3) has dimension $$2\ell + 1$$ (physicist tends to use j for spin/orbital angular momentum number).