I solved the Laplace equation in cylindrical coordinates with no $\theta$ dependence and got the solution $u(r,z) = \sum_{n=1}^\infty K_n \sinh (\mu_nz)J_0(\mu_nr)$.
I have the conditions: $u(r,1) = f(r)$ and $u(1,z) = 0$.
Using both of these conditions I get that:
$u(r,1) = \sum_{n=1}^\infty K_n \sinh (\mu_n)J_0(\mu_nr) = f(r)$
and
$u(1,z) = \sum_{n=1}^\infty K_n \sinh (\mu_nz)J_0(\mu_n) = \sum_{n=1}^\infty K_n \sinh (\mu_nz)J_0(0) = \sum_{n=1}^\infty K_n \sinh (\mu_nz)= 0$
This is because $\mu_n$ is a zero of $J_0$ and $J_0(0) = 1$.
How can I use these to obtain the coefficients $K_n$.
I feel like this would be easier if I was dealing with sin rather than sinh as then I would have a fourier sine series for which finding the coefficients is easy.
I also have the fact that $\int_0 ^1 J_0(\mu_mr)*J_0(\mu_nr)*r = \delta_{m,n}$. This is just the orthogonality of the eigenfunctions that come from the bessel equation.