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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 , where are certain constants such that and do not vanish simultaneously. If the straight line 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 are the upper half-plane , the lower half-plane , the left half-plane , the right half-plane , etc. The upper half-plane of the complex -plane can be mapped conformally (cf. Conformal mapping) onto the disc by the Möbius transformation

where is an arbitrary real number and .

How to Cite This Entry:
Half-plane. BSE-3 (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098