The Green's function for the 2D Helmholtz equation satisfies the following equation:


By Fourier transforming the Green's function and using the plane wave representation for the Dirac-delta function, it is fairly easy to show (using basic contour integration) that the 2D Green's function is given by


where $\operatorname{H}_0^{(1)}$ is the Hankel function of zeroth order and first kind.

However, this 2D Green's function diverges (logarithmically) at $\mathbf{r}=\mathbf{r}'$. Therefore, if we want it to be well-defined for $\mathbf{r}=\mathbf{r}'$, one can introduce a Gaussian cut-off function like so


where $a$ is some cut-off parameter.

Question: How do you evaluate this integral?

  • 1
    $\begingroup$ Schwinger's trick? $\endgroup$
    – Sunyam
    Commented May 14, 2019 at 10:38
  • $\begingroup$ The same basic contour integration should work here. $\endgroup$
    – Roger V.
    Commented Apr 29, 2020 at 6:14
  • $\begingroup$ How did you solve the integral in the first place? I doesn't seem to be that easy tbh.. $\endgroup$ Commented Sep 30, 2020 at 19:51

1 Answer 1


The poles are still in the same place so what is stopping you using residue theorem?

  • 1
    $\begingroup$ What contour do you have in mind? The gaussian piece is poorly behaved in the complex plane, as it diverges for $k=ix$ and $x\to \pm \infty$. $\endgroup$
    – user196574
    Commented Nov 29, 2022 at 17:56

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