I'm trying to solve the following problem:
Suppose that $u:\mathbb D \setminus\{0\} \to \mathbb R$ is harmonic and that $\lim_{z\to 0} u(z)=\infty$.
Show that $u$ can be written as $$u(z)=\beta \ln |z| + v(z) $$ where $\beta\in \mathbb R \setminus \{0\}$ and $v:\mathbb D \to \mathbb R $ is harmonic in $\mathbb D$ (Here $\mathbb D = \{|z|<1\} $).
I was also given the following hint:
Hint: Show that the residue of $(u_x - i u_y) $ is real.
I believe that the solution sould be related to Dirichlet's problem, and especially to Poisson's formula, but I don't understand how to use those to solve it.
Any help would will be appreciated!