I have the series

$\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$

and the boundary conditions

$\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n)=1.$

Here $J_\nu$ stands for a Bessel function and the coefficients $a_n$ are unknown. It can be readily found from the above BC at $r=0$ that

$a_0=1.$

A simple solution can be found for $\Phi=\pi/2$, namely

$a_0=1$ and $a_{n}=2\times(-1)^n$ for $n>1$.

However, I want to solve this problem for $\Phi=3\pi/4$. It is not even evident that a solution exists for $\Phi=3\pi/4$. Indeed, if one expands the Bessel functions into the Taylor series and equate to $0$ the coefficients before different powers of $x$ then one obtains that $a_n=0$ for all $n$. In any case, these problems seems to be ill-posed. I computed a numerical solution for this problem using different mehods (see e.g. Inverse problems for an asymptotic series which depends on a parameter) but still not sure that I did that well. Could someone direct me to a correct way?

Finally I need to compute the Laplace inverse for

$\Gamma(7/3)p^{-7/3}\psi(r,\theta;p)$

which is an electrostatic potential outside a rectangular corner of a homogeneous beam of charged particles provided that the so called Pierce electrodes compensate an electrostatic repulsion of the particles in the directions transversal to direction of the beam propagation.