Upper half-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 upper half-plane is defined as the set of complex numbers with strictly positive imaginary part, i.e.:

$\mathcal{H}:=\{x+iy\mid y>0\}$

Riemann mapping

The upper half-plane admits a particularly easy Riemann mapping; in fact, one coming from a fractional linear transformation:

$z\mapsto \frac{z-i}{z+i}$

Relation with other domains

• Right half-plane: The right half-plane maps bijectively to the upper half-plane via a rotation map $z\mapsto iz$
• Slit plane: The slit plane maps bijectively to the upper half-plane via the map $z\mapsto i\sqrt{z}$