Spherical Harmonics expansion of a Dirac Delta at the North Pole

Question

I think all the coefficients for the spherical harmonic expansion of a delta function at the north pole should be a constant (presumably 1), but I'm having difficulties calculating them. Could someone kindly take some time showing me how they are calculated, thanks a lot!

Answer 1

I don't see why they all have to be 1. A difficulty with the North pole is that the usual coordinate system has a singularity there. So think about a function that approximates the delta function, maybe something constant in a small radius around the North pole.

I'm using the definitions here: http://en.wikipedia.org/wiki/Spherical_harmonics#Orthogonality_and_normalization http://en.wikipedia.org/wiki/Spherical_coordinate_system (I think we are using the "physics" version of spherical coordinates.)

It seems to me that the $f_l^m$ coefficients should be zero whenever $m \ne 0$, because you are integrating $e^{im\varphi}$ from $\varphi=0$ to $2\pi$.