Thanks for your reply.
The general solution for potential outside the cylinder is than
\Phi(\rho>a, \varphi, z)=\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\big(J_{m}(k_{mn}\rho)+N_{m}(k_{mn}\rho)\big)\sinh(k_{mn}z)[A_{mn}\sin(m\varphi)+B_{mn}\cos(m\varphi)].
Is there sin hyp function in z dependence...
The potential on the side and the bottom of the cylinder is zero, while the top has a potential V_0. We want to find the potential outside the cylinder.
Can I use the same boundary conditions as for case of inside cylinder potential?
What is different?