Slit plane

This article defines a particular simply connected domain in $\mathbb {C}$, the complex numbers
View a complete list of particular simply connected domains

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.

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

More generally, the term slit plane is used for a generalized slit plane: the complement in $\mathbb {C}$ of any (not necessarily straight) ray (with endpoint) going off to infinity. Any such slit plane is contractible, though not necessarily star-like, and admits holomorphic logarithms and roots. More generally, any simply connected domain that is not the whole of $\mathbb {C}$ is contained in a generalized slit plane.

Riemann mapping

The slit plane admits an easy Riemann mapping -- in fact, it is a very special case of the Riemann mapping theorem. First, choose a holomorphic squareroot on the slit plane, with the property that the squraeroot of a positive real number is is positive squareroot. The image of the slit plane under this mapping is the right half-plane.

Then the Riemann mapping is given by:

$z\mapsto {\frac {{\sqrt {z}}-1}{{\sqrt {z}}+1}}$