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Difference between revisions of "Half-plane"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461701.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461702.png" /> are certain constants such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461704.png" /> do not vanish simultaneously. If the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461705.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461706.png" /> are the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461707.png" />, the lower half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461708.png" />, the left half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h0461709.png" />, the right half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h04617010.png" />, etc. The upper half-plane of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h04617011.png" />-plane can be mapped conformally (cf. [[Conformal mapping|Conformal mapping]]) onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h04617012.png" /> by the Möbius transformation
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h04617013.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h04617014.png" /> is an arbitrary real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046170/h04617015.png" />.
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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  $  Ax + By + C > 0 $,
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where $  A , B , C $
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are certain constants such that  $  A $
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and  $  B $
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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 $
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are the upper half-plane  $  y =  \mathop{\rm Im}  z > 0 $,
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the lower half-plane  $  y =  \mathop{\rm Im}  z < 0 $,
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the left half-plane  $  x = \mathop{\rm Re}  z < 0 $,
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the right half-plane  $  x = \mathop{\rm Re}  z > 0 $,
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etc. The upper half-plane of the complex  $  z $-
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plane can be mapped conformally (cf. [[Conformal mapping|Conformal mapping]]) onto the disc  $  | w | < 1 $
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by the Möbius transformation
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$$
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w  =  e ^ {i \theta }
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\frac{z - \beta }{z - \overline \beta \; }
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,
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$$
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where  $  \theta $
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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=16040
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article