Slit plane

Definition

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

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

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:

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

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

The slit plane also admits a holomorphic squareroot and holomorphic $n^{th}$ roots for higher $n$.