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Simple question: Why the below formula is right?(I want to know the proof)

The equivalent in 2D geometry is $$ \nabla^2 \ln \mid\vec\theta-\vec\theta'\mid = 2\pi\delta^2(\vec\theta-\vec\theta')\ ,$$ where $\delta^2$ is the 2D delta function.

Because I wondered above, I asked to stack exchange in astronomy, So I get the link which explains it(stack exchange in math1, stack exchange in math2)

But my major is not mathematics, so it is hard to understand.

Is it impossible to understand by using only the basic knowledge of mathematics? I know only basic calculus(from James Stewart's book) and some basic linear algebra.

There is no context related to this proof in the stack exchange, you tube, or any other website.

BAO
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  • Which chapter of Stewart is this? – Chee Han Dec 27 '22 at 06:25
  • @CheeHan Sorry, I edited my question properly. – BAO Dec 27 '22 at 06:26
  • Basic maths, yes and no, it depends (as always). If you only want to check the relation, your given links integrate the relation on a small disk around $\vec{\theta}'$ in order to check the property of Dirac delta on a test function; "basic" identities (such as Green's one) are then needed, but the calculations are quite messy. On the other hand, if you want to rederive the relation, one needs to find the Green function of the 2D Poisson equation, typically thanks to a Fourier transform (but the 2D case is special, due to problematic convergence). – Abezhiko Dec 27 '22 at 08:39

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