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 to the open unit disk; 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 $z\mapsto i{\sqrt {z}}$