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Laplace's equation in spherical coords 
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#1
May1009, 10:55 AM

P: 32

I have a simple question about the general solution to Laplace's equation in spherical coords.
The general solution is: [tex]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}[/tex] (where the [tex]a_{lm}, b_{lm}[/tex] coefficients can be found using the boundary conditions in question.) My problem lies in trying to understand the limits on the summation [tex]\sum^{l}_{m=l}[/tex]. Can anyone offer any help on this please? Thanks for reading, b. 


#2
May1409, 10:17 AM

P: 333

Probably due to the same reason why the irreducible representation of SO(3) has dimension
[tex]2\ell + 1[/tex] (physicist tends to use j for spin/orbital angular momentum number). 


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