Let $x,y\in \mathbb{R}^2$, $B_r(0)=\{x||x|\leq r\}$. Does the following function (denoted as $f(r)$) have a closed form expression?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
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4$\begingroup$ MSE is a right forum for such questions. $\endgroup$– user64494Commented Apr 24, 2021 at 17:08
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2$\begingroup$ For $r=1$ Mathematica answers $-\frac{\pi ^2}{4}$. $\endgroup$– user64494Commented Apr 24, 2021 at 17:41
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3$\begingroup$ A simple observation shows how $f(r)$ is related to $f(1)$. $\endgroup$– Anthony QuasCommented Apr 24, 2021 at 17:47
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2$\begingroup$ Hi wanjie. I've attempted to correct some grammatical errors in your question. I think your question is a good one, and most of the solution has been given in the comments, but I agree with others that it is probably more suitable for Mathematics Stack Exchange. MathOverflow is intended for questions about research-level mathematics, whereas Mathematics Stack Exchange is intended for more general mathematical questions such as this one. $\endgroup$– Tim CampionCommented Apr 24, 2021 at 18:03
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2$\begingroup$ @user64494 Make the simple substitution $x' = rx$, $y' = ry$. It's true that one does not get a straightforward scaling relation, but the result is simply that there is an extra $\log(r)$ term as seen in the general answer. $\endgroup$– Tim CampionCommented Apr 24, 2021 at 18:19
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