upper half-plane

In mathematics, the upper half-plane, $\,\mathcal\,,$ is the set of points in the Cartesian plane with > 0.

Complex plane

Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part:

:$\mathcal \equiv \ ~.$

The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the  axis is oriented vertically, the "upper half-plane" corresponds to the region above the  axis and thus complex numbers for which  > 0.

It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by  < 0, is equally good, but less used by convention. The open unit disk $\,\mathcal\,$ (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to $\,\mathcal\,$ (see "Poincaré metric"), meaning that it is usually possible to pass between $\,\mathcal\,$ and $\,\mathcal\; .$

It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.

The uniformization theorem for surfaces 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 of the upper half-plane and the real axis. It is the closure of the upper half-plane.

Affine geometry

The affine transformations of the upper half-plane include
# shifts (''x,y'') → (''x'' + ''c, y''), , and
# dilations (''x, y'') → (λ ''x'', λ ''y''), λ > 0.

Proposition: Let ''A'' and ''B'' be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes ''A'' to ''B''.
:Proof: First shift the center of ''A'' to (0,0). Then take λ = (diameter of ''B'')/(diameter of ''A'') and dilate. Then shift (0,0) to the center of ''B''.

Definition: $\mathcal \equiv \left\ ~.$

$\mathcal$ can be recognized as the circle of radius centered at (, 0), and as the polar plot of $\rho(\theta) = \cos \theta~.$

Proposition: (0,0), $\rho(\theta)$ in $\mathcal \,,$ and $(\,1, \tan \theta\,)$ are collinear points.

In fact, $\mathcal$ is the reflection of the line $\bigl\$ in the unit circle. Indeed, the diagonal from (0,0) to $(\,1, \tan \theta\,)$ has squared length $1 + \tan^2 \theta = \sec^2 \theta\, ,$ so that $\rho (\theta) = \cos \theta$ is the reciprocal of that length.

Metric geometry

The distance between any two points and in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from to either intersects the boundary or is parallel to it. In the latter case and 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 and 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 $\mathcal \;.$ Distances on $\mathcal$ can be defined using the correspondence with points on $\bigl\$ and logarithmic measure on this ray. In consequence, the upper half-plane becomes a metric space. The generic name of this metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model.

Generalizations

One natural generalization in differential geometry is hyperbolic -space $\, \mathcal^n \, ,$ the maximally symmetric, simply connected, -dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is $\, \mathcal^2 \,$ since it has real dimension 2.

In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product $\, \mathcal^n \,$ of copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space $\, \mathcal_n \,,$ which is the domain of Siegel modular forms.

See also

* Cusp neighborhood
* Extended complex upper-half plane
* Fuchsian group
* Fundamental domain
* 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

Complex analysis
Hyperbolic geometry
Differential geometry
Number theory
Modular forms

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