Branch of the logarithm

Remarks

Each branch $L$ of the logarithm on $U$ is holomorphic on $U$, with $L'(z) = 1/z$. If $U$ is a domain and $L_1, L_2$ are two branches on $U$, then $L_1 - L_2$ is the constant $2 \pi i k$ for one $k \in \mathbb{Z}$. There is no branch of the logarithm on $\mathbb{C} \setminus \{0\}$, or on any open set that contains a circle with center $0$. The principal branch $\operatorname{Log}$ is a branch on $\mathbb{C} \setminus (-\infty, 0]$. The functions $\operatorname{Log} z + 2 \pi i k$ with $k \in \mathbb{Z}$ are the other branches on this set.