Laplace's equation in spherical co-ords

[benabean]

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.

 

[matematikawan]

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).