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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.