There is a cylinder with radius $\rho_o$ and height $h$. The lids are on the planes $z=0$ and $z=h$.
$\nabla ^2 \phi = 0$ , $\phi = \phi_o$ on the upper lid, $\phi=0$ every where else on the cylinder. $\phi$ is finite.
What I have done:
There is cylindrical symmetry so no there are only radial and z dependencies.
$$\rho^2R''(\rho)+\rho R'(\rho)+\frac{Z''(z)}{Z} \rho^2 R(\rho)=0$$
$$R(\rho)=J_0(\frac{j_{0n}}{\rho_0}\rho)$$
$j_{0n}$ is the $n^{th}$ zero of the bessel function.
$$Z(z)=\sinh(\frac{j_{0n}}{\rho_0}z)$$
$$\phi(\rho,z)=\sum^{\infty}_{n=1}C_n \sinh(\frac{j_{0n}}{\rho_0}z)J_0(\frac{j_{0n}}{\rho_0}\rho)$$
The boundary condition at $z=h$
$$\phi_0=\sum^{\infty}_{n=1}C_n \sinh(\frac{j_{0n}}{\rho_0}h)J_0(\frac{j_{0n}}{\rho_0}\rho)$$
How do I find the coefficients $C_n$. This looks like "Hankel series coefficients", which I doubt is a concept that exists.
EDIT: I tried to do some stuff and this is the plot of
$$\sum ^{900} _ {n=0}J_1( j_{0n})J_0(j_{0n}\rho)$$
So it looks somewhat good, but i'd like to know more.
using this orthogonality property, I was able to decompose into discrete coefficients
http://www.hit.ac.il/staff/benzionS/Differential.Equations/Orthogonality_of_Bessel_functions.htm
$$\phi_0 = \sum^{\infty}_{n=1}C_n J_0(j_{0n}\rho)$$
$$\int ^1 _ 0 \phi_0 \rho J_0(j_{0k}\rho)d\rho= \sum^{\infty}_{n=1}C_n \int ^1 _ 0 J_0(j_{0n}\rho) \rho J_0(j_{0k}\rho)d\rho$$
$$\phi_0 \frac{J_1(j_{0k})}{j_{0k}}=\sum^{\infty}_{n=1}C_n \delta_{nm}\frac12J_1^2(j_{0k })=C_k \frac12J_1^2(j_{0k })$$
And this is the numerical result up to k=100