I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true:

$$\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \log{\left(x^2+y^2\right)^{\frac{1}{2}}} = 2\pi\delta(x)\delta(y).$$

Function $\log$ is natural logarithm, i.e. $\log e=1$, and $\delta(x)$ and $\delta(y)$ are Dirac's delta "functions". Unfortunately, I cant remember where in the literature I have seen this equation since the last time I was using it was in 2005. For that reason, I kindly ask if someone can point me to the papers or books which I can use for reference.

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    $\begingroup$ Would this be better suited on Mathematics (assuming they don't get too upset about the Dirac Delta Function)? Have you tried actually evaluating that expression? I am not sure it gives DDFs $\endgroup$ – Aaron Stevens Jan 22 '19 at 21:23
  • $\begingroup$ Yes probably, I wasn't even aware that group exists since I was so focused on explaining some numerical methods in physics. Will try to ask question there, thanks for advice $\endgroup$ – gallieo1985 Jan 22 '19 at 21:29
  • $\begingroup$ Math mods: Please merge. $\endgroup$ – Qmechanic Jan 22 '19 at 22:50

The expression mathematically is saying that, the natural log function is the Green's function for free-space Laplace equation in 2D. (in 3D it will be the Coulomb 1/r potential)

This is a pretty standard and well-known thing, and be found in almost every standard books either on linear Partial Differential Equations, or Potential Theory. You can also find it in wikipedia: https://en.wikipedia.org/wiki/Green%27s_function

You may check out a few references there. But to be honest, I do not think it is necessary to cite anything. I am in the math department though.

  • $\begingroup$ Yes I know its standard, however I am having trouble finding citable reference in standard books $\endgroup$ – gallieo1985 Jan 23 '19 at 0:33
  • $\begingroup$ I personally will suggest: Kellogg, Foundations of Potential Theory. This is a pretty good book, and there is a whole section about The Logarithmic Potential. $\endgroup$ – Zecheng Gan Jan 23 '19 at 1:07

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