# Integrating log |z-w| over a disk in the complex plane.

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: $$$$u_{r}=\left(\chi_{r}-\delta_{0}\right) * E$$$$ where $$\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.