Let $D$ be any disk centered at $1$ and not containing $0$. Prove that $(\log, D)$ can be analytically continued along any curve $\gamma$ in $\mathbb C \setminus \{ 0 \}$. Here $\log$ is the principal branch of $\log$ function.
I know that any analytic function can be moved along an analytic covering map, so we just need to find such map?