 Solution

Use separation of variables, writing $\Psi =F(r)\Theta (\theta )$. After multiplication by $r^2/(F\Theta )$, the radial and angular terms are functions of independent variables, so setWith $x=\cos \theta$, the angular equation isRegularity at both poles selects $\lambda =\ell (\ell +1)$, $\ell =0,1,\dots$, and the Legendre polynomials $P_{\ell }(x)$; the other angular branch is singular at a pole. The radial Euler-Cauchy equation is $r^2{F}^{\prime \prime }+2r{F}^{\prime }-\ell (\ell +1)F=0$. A power $r^s$ gives $(s-\ell )(s+\ell +1)=0$, hence $F=a_{\ell }{r}^{\ell }+b_{\ell }{r}^{-\ell -1}$.

Superposing these axisymmetric spherical harmonics givesThe coefficients must give convergence appropriate to the domain. In a region containing the origin, regularity removes the negative powers; in an exterior region with zero potential at infinity, it removes the growing powers and the constant. No such radial restriction is needed for a general annular region.