Difference between revisions of "Half-plane"
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| + | $#C+1 = 15 : ~/encyclopedia/old_files/data/H046/H.0406170 Half\AAhplane | ||
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| − | where < | + | 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|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 $. | ||
Latest revision as of 19:42, 5 June 2020
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
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