I am trying to find the solution of $ \nabla^2U = U_{rr}+\frac{1}{r}U_r+U_{zz}=0$, where $0\leq r <\infty$, and $z>0$. The boundary conditions are given by $U(r,0)=A$ (A is constant), for $0\leq r <a$, and $U(r,0)=0$ , for $r >a$.
Let $H(U(r,z)) = u(\alpha,z)$, $H(f(r)) = f(\alpha)$, then we have $$ f(\alpha) = -\alpha^2u+u_{zz}=0 $$
Then I tried to solve this differential equation for $u$, and am planning to apply the inverse Hankel transform. My solution to this equation is $$ U = C_1e^{\alpha z}+C_2e^{-\alpha z} $$ Is that look correct? Also, to apply the inverse Hankel transform, is there a way I can figure out the $n$ in the Bessel function $J_n$? I am still confused about how to solve for $U$. Thanks:)