Branch of the logarithm

Definition

Let $U$ be an open subset of $\mathbb{C} \setminus \{0\}$. A branch of the logarithm on $U$ is a continuous function $L : U \to \mathbb{C}$ such that $e^{L(z)} = z$ for each $z \in U$. Thus $L(z)$ is a complex logarithm of $z$ for each $z \in U$.

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.

Explanations

L1 human

The logarithm has many values at each point, so to get a function we must choose one value at each point, and the choice must change continuously. Such a choice is a branch. If you go once around $0$, the argument grows by $2\pi$, so no continuous choice is possible on a full circle around $0$.