I'm reading this. Equation (522) gives the general solution to the Laplace equation. What I'm stuck about, is how to determine the coefficients $a_m$, $\beta_m$, and $\theta_m$ for non-trivial boundary conditions.
For instance, setting $r=1$ in the equation for $\psi$ works. But then I don't know how to determine $\theta_m$.
In particular, how are the coefficients determined for boundary conditions, of the most general kind possible? For example, flow around a circle or a wedge? The site I reference doesn't really elaborate on how the coefficients are obtained for these cases.