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| {{refimprove|date=February 2010}}
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| In [[mathematics]], the '''upper half-plane''' '''H''' is the set of [[complex number]]s with positive [[imaginary part]]:
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| :<math>\mathbb{H} = \{x + iy \;| y > 0; x, y \in \mathbb{R} \}.</math>
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| The term arises from a common visualization of the complex number ''x + iy'' as the point ''(x,y)'' in [[the plane]] endowed with [[Cartesian coordinates]]. When the [[Y-axis]] is oriented vertically, the "upper [[half-plane]]" corresponds to the region above the X-axis and thus complex numbers for which ''y'' > 0.
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| It is the [[domain (mathematics)|domain]] of many functions of interest in [[complex analysis]], especially [[modular form]]s. The lower half-plane, defined by ''y'' < 0, is equally good, but less used by convention. The [[open unit disk]] '''D''' (the set of all complex numbers of [[absolute value]] less than one) is equivalent by a [[conformal mapping]] to '''H''' (see "[[Poincaré metric]]"), meaning that it is usually possible to pass between '''H''' and '''D'''.
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| It also plays an important role in [[hyperbolic geometry]], where the [[Poincaré half-plane model]] provides a way of examining [[hyperbolic motion]]s. The Poincaré metric provides a hyperbolic [[metric tensor|metric]] on the space.
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| The [[uniformization theorem]] for [[surface]]s states that the '''upper half-plane''' is the [[covering map#Universal covers|universal covering space]] of surfaces with constant negative [[Gaussian curvature]].
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| ==Generalizations==
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| One natural generalization in [[differential geometry]] is [[hyperbolic space|hyperbolic ''n''-space]] '''H'''<sup>''n''</sup>, the maximally symmetric, [[simply connected]], ''n''-dimensional [[Riemannian manifold]] with constant [[sectional curvature]] −1. In this terminology, the upper half-plane is '''H'''<sup>2</sup> since it has [[real (numbers)|real]] [[dimension]] 2.
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| In [[number theory]], the theory of [[Hilbert modular form]]s is concerned with the study of certain functions on the direct product '''H'''<sup>''n''</sup> of ''n'' copies of the upper half-plane. Yet another space interesting to number theorists is the [[Siegel upper half-space]] '''H'''<sub>''n''</sub>, which is the [[domain (mathematics)|domain]] of [[Siegel modular form]]s.
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| ==See also==
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| * [[Cusp neighborhood]]
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| * [[Extended complex upper-half plane]]
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| * [[Fuchsian group]]
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| * [[Fundamental domain]]
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| * [[Hyperbolic geometry]]
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| * [[Kleinian group]]
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| * [[Modular group]]
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| * [[Riemann surface]]
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| * [[Schwarz-Ahlfors-Pick theorem]]
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| ==References==
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| *{{MathWorld|title=Upper Half-Plane|urlname=UpperHalf-Plane}}
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| [[Category:Complex analysis]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Differential geometry]]
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| [[Category:Number theory]]
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| [[Category:Modular forms]]
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| [[de:Obere Halbebene]]
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| [[it:Semipiano]]
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