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I've come to realize that its somehow harder to find results for this equation than for the three-dimensional one.

For example the wikipedia article on Green's functions has a list of green functions where the Green's function for both the two and three dimensional Laplace equation appear. Also the Green's function for the three-dimensional Helmholtz equation but nothing about the two-dimensional one.

The same happens in the Sommerfield condition article. The condition is written in a generic fashion, depending on the number of dimensions $n$, but when it's time to show an example showing the solution of a point source only the three-dimensional case is shown.

Is it just a coincidence? or the solution to the point source in the two dimensional case: $$ G\left(\mathbf{x},\mathbf{y}\right)=\frac{i}{4}H_0^1\left(\kappa\left\vert\mathbf{x}-\mathbf{y}\right\vert\right) $$ is not a true Green's functions for some reason?

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    $\begingroup$ What is a "true" Green's function? The difference is that in more than two dimensions there is a solution of $\Delta u=\delta$ which vanishes at infinity, in two or one dimension there is not. $\endgroup$
    – Michael Renardy
    Commented May 11, 2022 at 3:40
  • $\begingroup$ That was part of my question, that maybe for some reason I don't know this function I show was not considered a Green function. However, I don't know that you mean by your difference. The article in the wikipedia does indeed show a green function for the two-dimensional laplacian (which, as you say, does not vanish at infinity). But my question is why the two-dimensional Helmholtz equation is missing? $\endgroup$
    – Manuel Pena
    Commented May 11, 2022 at 7:57
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    $\begingroup$ The Helmholtz equations stems from separating variables in the wave equation. Physicists are well aware that wave propagation in 2d is very different from 3d. Living in a two dimensional world we would be begging to go deaf. $\endgroup$
    – Kurt G.
    Commented May 11, 2022 at 9:21
  • $\begingroup$ Green's function for the two-dimensional Helmholtz equation is derived in this work: sbfisica.org.br/rbef/pdf/351304.pdf $\endgroup$
    – Ivo Ita
    Commented Feb 7, 2023 at 15:01

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You seek the solution of $$(\nabla^2+\kappa^2+i\epsilon)G(\mathbf{r})=\delta(\mathbf{r}),$$ in the limit $\epsilon\rightarrow 0^+$, which is given by a Hankel function of the first kind, $$G(\mathbf{r})=\lim_{\epsilon\rightarrow 0^+}\int\frac{d^2\mathbf{k}}{(2\pi)^2}e^{i\mathbf{k}\cdot\mathbf{r}}\frac{1}{\kappa^2+i\epsilon-k^2}=\frac{1}{4i}H_0(\kappa r).$$ There is a logarithmic singularity at $r=0$, but it's a valid Green function.

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