I got stuck trying to prove that if $ \Omega$ is an open connected subset of $\mathbb{C}$ and $\varphi, \psi : \Omega \rightarrow \mathbb{C}$ are two holomorphic logarithmic branches, then for each $z \in \Omega $ exists a $n \in \mathbb{Z}$ such that $\varphi(z) = \psi(z)+ 2 \pi i n $
Any help ?
Even more is true: The exists a $n \in \mathbb{Z}$ such that $\varphi(z) = \psi(z)+ 2 \pi i n $ for all $z \in \Omega $ (i.e. the $n$ is independent of $z$).
If $\varphi, \psi : \Omega \rightarrow \mathbb{C}$ are holomorphic branches of the logarithm then $$ e^{\varphi(z) - \psi(z)} = \frac{ e^{\varphi(z)}}{e^{\psi(z)}} = \frac{z}{z} = 1 \implies \frac{1}{2 \pi} \bigl(\varphi(z) - \psi(z) \bigr) \in \Bbb Z. $$ for all $z \in \Omega $. Now use that a continuous function on a connected set which takes only integer values is necessarily constant.