View source for Upper half-plane

{{Short description|Complex numbers with non-negative imaginary part}}
In [[Mathematics|mathematics]], the '''upper half-plane''', <math>\,\mathcal{H}\,</math>, is the set of points <math>(x,y)</math> in the Cartesian plane with 
<math>y > 0</math>. The '''lower half-plane''' is defined similarly, by requiring that <math>y</math> be negative instead. Each is an example of two-dimensional [[Half-space (geometry)|half-space]].

==Affine geometry==
The [[Affine transformation|affine transformation]]s of the upper half-plane include
# shifts <math>(x,y)\mapsto (x+c,y)</math>, <math>c\in\mathbb{R}</math>, and
# dilations <math>(x,y)\mapsto (\lambda x,\lambda y)</math>, <math>\lambda > 0 </math>.

'''Proposition:''' Let <math>A</math> and <math>B</math> be [[Semicircle|semicircle]]s in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes 
<math>A</math> to <math>B</math>.
:Proof: First shift the center of <math>A</math> to <math>(0,0)</math>. Then take <math>\lambda=(\text{diameter of}\ B)/(\text{diameter of}\ A)</math>   
and dilate. Then shift <math>(0,0)</math> the center of <math>B</math>.

'''Definition:''' <math>\mathcal{Z} := \left\{\left( \cos^2(\theta),\tfrac{1}{2} \sin(2\theta) \right) \mid 0 < \theta < \pi \right\} </math>.

<math>\mathcal{Z}</math> can be recognized as the circle of radius <math>1/2</math> centered at <math>(1/2,0)</math>, and as the polar plot of 
<math>\rho(\theta) = \cos(\theta)</math>.

'''Proposition:''' <math>(0,0)</math>, <math>\rho(\theta) \in \mathcal{Z}</math>, and <math>(1,\tan(\theta))</math> are collinear points.

In fact, <math>\mathcal{Z}</math> is the reflection of the line <math>\bigl\{(1, y) \mid y > 0  \bigr\}</math> in the [[Unit circle|unit circle]]. Indeed, the diagonal from 
<math>(0,0)</math> to <math>(1, \tan(\theta))</math> has squared length <math>1 + \tan^2(\theta) = \sec^2(\theta) </math>, so that <math>\rho(\theta) = \cos(\theta)</math> 
is the reciprocal of that length.

==Metric geometry==
The distance between any two points <math>p</math> and <math>q</math> in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from <math>p</math> to <math>q</math> either intersects the boundary or is parallel to it. In the latter case <math>p</math> and <math>q</math> lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation. In the former case <math>p</math> and <math>q</math> lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to 
<math>\mathcal{Z} </math>. Distances on <math>\mathcal{Z} </math> can be defined using the correspondence with points on <math>\bigl\{(1, y)\mid y > 0 \bigr\}</math> and logarithmic measure on this ray. In consequence, the upper half-plane becomes a [[Metric space|metric space]]. The generic name of this metric space is the hyperbolic plane. In terms of the models of [[Hyperbolic geometry|hyperbolic geometry]], this model is frequently designated the [[Poincaré half-plane model]].

==Complex plane==
Mathematicians sometimes identify the Cartesian plane with the [[Complex plane|complex plane]], and then the upper half-plane corresponds to the set of [[Complex number|complex number]]s with positive imaginary part:

:<math>\mathcal{H} := \{x + iy \mid y > 0;\ x, y \in \mathbb{R} \} .</math>

The term arises from a common visualization of the complex number <math>x+iy</math> as the point <math>(x,y)</math> in [[Plane (geometry)|the plane]] endowed with Cartesian coordinates. When the <math>y</math> axis is oriented vertically, the "upper half-plane" corresponds to the region above the <math>x</math> axis and thus complex numbers for which <math>y > 0</math>.

It is the [[Domain of a function|domain]] of many functions of interest in [[Complex analysis|complex analysis]], especially [[Modular form|modular form]]s. The lower half-plane, defined by
<math>y < 0</math> is equally good, but less used by convention. The open unit disk <math>\mathcal{D}</math> (the set of all complex numbers of [[Absolute value|absolute value]] less than one) is equivalent by a conformal mapping to <math>\mathcal{H}</math> (see "[[Poincaré metric]]"), meaning that it is usually possible to pass between 
<math>\mathcal{H}</math> and <math>\mathcal{D}</math>.

It also plays an important role in [[Hyperbolic geometry|hyperbolic geometry]], where the [[Poincaré half-plane model]] provides a way of examining [[Hyperbolic motion|hyperbolic motion]]s. The Poincaré metric provides a hyperbolic [[Physics:Metric tensor|metric]] on the space.

The [[Uniformization theorem|uniformization theorem]] for [[Surface (topology)|surface]]s states that the '''upper half-plane''' is the universal covering space of surfaces with constant negative [[Gaussian curvature]].

The '''closed upper half-plane''' is the [[Union (set theory)|union]] of the upper half-plane and the real axis. It is the [[Closure (topology)|closure]] of the upper half-plane.

==Generalizations==
One natural generalization in [[Differential geometry|differential geometry]] is [[Hyperbolic space|hyperbolic <math>n</math>-space]] <math>\mathcal{H}^n</math>, the maximally symmetric, simply connected, <math>n</math>-dimensional [[Riemannian manifold]] with constant [[Sectional curvature|sectional curvature]] <math>-1</math>. In this terminology, the upper half-plane is 
<math>\mathcal{H}^2</math> since it has real [[Physics:Dimension|dimension]] <math>2</math>.

In [[Number theory|number theory]], the theory of [[Hilbert modular form]]s is concerned with the study of certain functions on the direct product <math>\mathcal{H}^n</math> of 
<math>n</math> copies of the upper half-plane. Yet another space interesting to number theorists is the [[Siegel upper half-space]] <math>\mathcal{H}_n</math>,
which is the domain of [[Siegel modular form]]s.

==See also==
* [[Cusp neighborhood]]
* Extended complex upper-half plane
* [[Fuchsian group]]
* [[Fundamental domain]]
* [[Half-space (geometry)|Half-space]]
* [[Kleinian group]]
* [[Modular group]]
* [[Riemann surface]]
* [[Schwarz–Ahlfors–Pick theorem]]
*[[Moduli stack of elliptic curves]]

==References==
*{{MathWorld|title=Upper Half-Plane|urlname=UpperHalf-Plane}}

[[Category:Complex analysis]]
[[Category:Hyperbolic geometry]]
[[Category:Differential geometry]]
[[Category:Number theory]]
[[Category:Modular forms]]

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it:Semipiano

{{Sourceattribution|Upper half-plane}}