Half-plane

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 $.