Let's suppose I want to solve Laplace's equation in Axial Symmetry:
for some function $\psi=\psi(\rho,z)$, where $\rho\in[0,\infty)$ and $z\in(-\infty,\infty)$. There are quite a few examples of how to solve Laplace's equation using
DSolve in the documentation, but as it seems, only for finite regions or with finitely-placed boundaries. But there are solutions (very important in physics) for example
for some constants $(m,l)$. The first describes a point source, and the second describes a finite line source.
I am interested in how one might obtain these solutions from
DSolve by inputting Laplace's equation and appropriate boundary conditions. One would need to set a boundary at infinity (as in an infinite Euclidean distance from the source) where ($\psi=0$) and then somehow adding a boundary where the source is located ($\psi=-\infty$).
I am interested how this may be done in general, not just for Laplace's equation, so making variable substitutions may not always be possible I guess.
Any help is appreciated.