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Let's suppose I want to solve Laplace's equation in Axial Symmetry: $$ \nabla^2\psi=\partial^2_{\rho}\psi+\partial^2_{z}\psi+\frac{1}{\rho}\partial_{\rho}\psi=0, $$ 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 $$ \psi=-\frac{m}{\sqrt{\rho^2+z^2}}\,,\quad\psi=\frac{1}{2}\ln\left(\frac{\sqrt{\rho^2+(z-l)^2}+\sqrt{\rho^2+(z+l)^2}-2l}{\sqrt{\rho^2+(z-l)^2}+\sqrt{\rho^2+(z+l)^2}+2l}\right) $$ 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.

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  • $\begingroup$ You mentioned that you want a boundary condition at infinity, but you did not specify is $\rho$ goes to infinity, or $z$ or both. $\endgroup$ Jan 14 at 21:33
  • $\begingroup$ Here $\rho$ ranges from 0 to $\infty$ and $z$ from $-\infty$ to $\infty$. When I say that the boundary is ``at infinity'', I mean at any infinite Euclidean distance away from the source. $\endgroup$ Jan 14 at 22:11
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    $\begingroup$ Perhaps you should update your post with these details, so people have the full picture $\endgroup$ Jan 14 at 22:28
  • $\begingroup$ GreenFunction? $\endgroup$
    – xzczd
    Jan 15 at 3:57

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