Slit plane

Definition

The slit plane is defined as the following open subset of $\mathbb {C}$:

$\mathbb {C} \setminus \{z\in \mathbb {R} \mid z\leq 0\}$

In other words, it is the complement in $\mathbb {C}$ of the half-line of nonpositive reals.

The slit plane is a star-like domain, with 1 as a star point. In particular, it is simply connected, and admits a holomorphic logarithm, given by:

$re^{i\theta }\mapsto \log r+i\theta$

where $\theta \in (-\pi ,\pi )$ is the principal argument.