Harmonic functions and simple connectivity

Question

I have a problem in complex analysis.

I want to prove that : A region(open, connected) $G \in \mathbb C$ is simply connected if and only if each harmonic function $u$ on $G$ is a real part of an analytic function $f $

I proved the "only if" part. But I cannot do anything for the "if" part. Any hint or proof will be appreciated.

Thank you.

Answer 1

Hint: Suppose $G$ is not simply connected, then there exists a closed curve $\gamma:I\rightarrow G$ and $z_0 \in G^c$ such that the winding number $\omega(\gamma, z_0) \neq 0$. Look at the function \begin{align} u(z)=\log|z-z_0| \end{align} which is harmonic on $G$ but $u(z)$ doesn't have a harmonic conjugate on $G$.