Half plane

In Euclidean geometry , a straight line divides a plane into two half-planes . If you add the straight line to one of the half-planes, one speaks of a closed half-plane , a half-plane without the straight line is called an open half-plane .

Upper half plane

upper half plane

The plane of complex numbers (and also ) is divided into two half-planes by any straight line. If this straight line is identical with the real numbers (or with the x-axis ), the set of complex numbers with a positive imaginary part is called the upper half-plane and is called within the function theory as (which in other contexts often for quaternions stands). ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {H} }$

${\displaystyle \mathbb {H} =\{x+iy\in \mathbb {C} \mid y>0\}}$.

It is the domain of several interesting functions such as B. Dedekind's η-function and plays an important role in modular forms and elliptical curves over complex numbers . The set of holomorphic functions on the upper half-plane , which are suitably bounded , form a Hardy space . is an unrestricted, simply connected area which can be mapped biholomorphically onto the unit disk (see also Riemann mapping theorem ). The lower half-plane could also be viewed analogously, since it has the same properties. ${\displaystyle \mathbb {H} }$

Generalizations

In general, the half-plane is a special half-space . In synthetic geometry , half-planes and half-spaces are defined by a division function, whereby more general coordinate ranges can be used instead of real numbers .

An important generalization is the Siegel half-space .

Publications

• Eberhard Freitag, Rolf Busam: Funktionentheorie 1 , 4th edition, Springer, Berlin (2006), ISBN 3-540-31764-3
• Max Koecher , Aloys Krieg : Elliptical functions and modular forms , 2nd edition, Springer, Berlin (2007) ISBN 978-3-540-49324-2