How do I find the distribution of the laplacian operator acting on Log |f|

Question

Can someone give me some ideas/insight/suggestions on approaching this problem:

Calculate the distribution $u(x) = \Delta \log{|\,f\,|}$ where $f$ is a meromorphic function that doesn't vanish identically. $f$ lies in an open set $\Omega \subset \mathbb{R}^2$. Here $\Delta$ is the Laplace operator.

Thanks in advance!

Answer 1

Hint: By the factorization theorem we can write $$f(z)=g(z)\prod_i (z-a_i)^{m_i},$$ for some non-vanishing holomorphic function $g$, $a_i\in \Omega$ and $m_i\in \mathbb{Z}$. Then $$ \ln |f(z)| = \ln|g(z)|+\sum_i m_i\ln|z-a_i|. $$ Now just recall that $\ln|z|$ (modulo some constant) is the fundamental solution of the laplacian.