Half-plane
From Encyclopedia of Mathematics
Revision as of 19:42, 5 June 2020 by Ulf Rehmann (talk | contribs) (tex encoded by computer)
The set of points in a plane situated to one side of a given straight line in that plane. The coordinates of the points of a half-plane satisfy an inequality $ Ax + By + C > 0 $, where $ A , B , C $ are certain constants such that $ A $ and $ B $ do not vanish simultaneously. If the straight line $ Ax + By + C = 0 $ itself (the boundary of the half-plane) belongs to the half-plane, the latter is said to be closed. Special half-planes on the complex plane $ z = x + iy $ are the upper half-plane $ y = \mathop{\rm Im} z > 0 $, the lower half-plane $ y = \mathop{\rm Im} z < 0 $, the left half-plane $ x = \mathop{\rm Re} z < 0 $, the right half-plane $ x = \mathop{\rm Re} z > 0 $, etc. The upper half-plane of the complex $ z $- plane can be mapped conformally (cf. Conformal mapping) onto the disc $ | w | < 1 $ by the Möbius transformation
$$ w = e ^ {i \theta } \frac{z - \beta }{z - \overline \beta \; } , $$
where $ \theta $ is an arbitrary real number and $ \mathop{\rm Im} \beta > 0 $.
How to Cite This Entry:
Half-plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Half-plane&oldid=47162
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Half-plane&oldid=47162"
TeX auto
TeX done