I would like to get a step-by-step solution of the Poisson equation (in polar coordinates) below
$$\nabla^2\psi(r, \phi) = 2 k(r, \phi)$$
where $\psi(r, \phi)$ is seperable (i.e. $\psi(r, \phi) = f(r)g(\phi)$) and
$$k(r, \phi) = \frac{\theta_E}{2r} \left( 1 + k_m \cos(m \phi) \right)$$
with $\theta_E, k_m, m$ being constants.
The solution for $\psi(r, \phi)$ is given in this paper, https://iopscience.iop.org/article/10.1088/0004-637X/765/2/134/pdf (Eq. 4-5, for $m\ne1$ and $m=1$, respectively). The solution given in the above paper is,
$$\psi(r, \phi) = \theta_E~ r \left( 1 - \frac{k_m}{m^2 - 1} \cos(m\phi) \right)\\m\ne1$$
and
$$\psi(r, \phi) = \theta_E~ r \left( 1 + \frac{k_m}{m^2 - 1} \ln (r) \cos\phi \right) \\m=1$$
Any help is welcome.