I found the following expression in IMRN International Mathematics Research Notices 2005, No. 34, page# 2064.
Let $\phi$ be a $C_c^{\infty}(\mathbb{C})$ then $$\int_{\mathbb{C}}\frac{1}{2\pi}\log|z|\Delta\phi(z)dxdy = \phi(0).$$ Here $\Delta \phi$ is the Laplacian of $\phi$.
I am not entirely sure how to get this integral. Any hint or explanation will be a great help.
So, it essentially says that the $\textit{distribution}~$ Laplacian of $\displaystyle -\frac{1}{2\pi}\log\left( \frac{1}{|z|}\right)$ with respect to $z$ is the point mass at $0$. For more details, one can see Theodore W. Gamelin's book on "Complex Analysis, Page # 393-394"[1]. It's a beautiful application of Green's second identity.
. [1]: https://www.springer.com/gp/book/9780387950938