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I am trying to drive the integral form relation between pair distribution function and structure factor in 2 dimensions. In 3D we get:

enter image description here

Where enter image description here

What would be the answer for $g(r)$ in 2D, my answer is:

$$g(r)-1=1/(\rho_0(2\pi)^2)\int{2\pi q(S(q)-1)\sin(qr)/(qr) dq}$$

$$S(q)-1=\rho_o\int{(2\pi r) (g(r)-1) \sin(qr)/(qr) dr}$$

but I think my answer is wrong as I'm calculating the pair distribution function for my simulation results from the formula I derived and the answer is not what I expect. I know the other formulas of $g(r)$ which are more straightforward, I am just doing it as an exercise.

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So, I found the answer to my question, in the following article eq(3) is giving the answer for $n=0$. My answers were correct too.

https://arxiv.org/abs/1605.06542

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