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?