# Is there a way to know the non-zero coefficient of $\cot(\theta)$ expansion in spherical harmonics?

I'm currently trying to find an analytical solution to the Poisson equation for a given distribution using a multipole expansion. During this task, I found the radial expansion and everything else, but I'm strunggling to find if there is a good way to expand $$\cot(\theta)$$ in spherical harmonics ($$Y_l^0$$, because I have azimuthal symmetry). Given the form of the function $$\cot$$, I know that the series will be infinite, but i don't know wich terms are zero.

Does anybody know a handbook where this can be found? Or if there is a way to know which coefficients are zero?

• If cotg means cotangent, it is usually written cot. – G. Smith Aug 2 '19 at 17:34
• What's wrong with the usual method of computing $\int Y^*_{lm} \cot$? – Brick Aug 2 '19 at 18:35
• The series is infinite, because cot(theta) goes to infinity at some points. And depending on the coefficents i get a different functional form, so it will be nice to know the coefficents that are importan or at least a way to know when i can truncate the series. – Nuclear Mati Aug 2 '19 at 18:52
• The series is usually infinite, so it still seems like the usual techniques should work. Since you only need $m=0$ you should quickly find that you only need either even or odd values of $l$ too since the parity of the spherical harmonics will go with the parity of $l$ in this case. We wouldn't be able to tell you where to truncate anyway since that depends on how much error you can tolerate and what other approximations you have. Like usually the radial expansion scales somehow with the angular one. – Brick Aug 2 '19 at 21:22
• You can calculate the coefficients one-by-one with Mathematica or something, and see when you get something accurate enough. – Javier Aug 2 '19 at 22:19