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Let $ E=\frac{1}{\pi} \log |z|^{2} $ and $ \chi_{r}=\frac{1}{\pi r^{2}} \chi_{\Delta(0, r)} $ where $\chi_{\Delta(0, r)}$ is the characteristic function of the disk with radius $r$ centered at $0$, both functions defined in $\mathbb{C}$.

I would like to compute explicitly the following: \begin{equation} u_{r}=\left(\chi_{r}-\delta_{0}\right) * E \end{equation} where $\delta_{0}$ is the Dirac delta function.

So far I understand that $u_{r}(z)=0 \text { in }|z|>r$ as the function $E$ is harmonic there, so $E(z)$ is equal to its average, but computing the explicit formula when $|z|\leq r$ is a different issue.

The answer should be: $$ u_{r}=\frac{1}{\pi}\left(\frac{|z|^{2}}{r^{2}}-1+\log \left(\frac{r^{2}}{|z|^{2}}\right)\right) $$

I've found in the page 7 of these notes https://arxiv.org/pdf/0804.4689.pdf something that could be useful, but still I'm a bit confused of how to derive it from its definition.

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