Upper half-plane

In mathematics, the upper half-plane, $\,{\mathcal {H}}\,,$ is the set of points $(x, y)$ in the Cartesian plane with $y$ > 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 {H}}\equiv \{x+iy\mid y>0;x,y\in \mathbb {R} \}~.

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.

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

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), $c \in R$ , 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: ${\mathcal {Z}}\equiv \left\{\,\left(\cos ^{2}\theta \,,\;{\tfrac {1}{2}}\sin 2\theta \,\right)\;|\;0<\theta <\pi \,\right\}~.$

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

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

In fact, ${\mathcal {Z}}$ is the reflection of the line ${\bigl \{}(\,1,y\,)\mid y>0{\bigr \}}$ 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 $p$ and $q$ in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from $p$ to $q$ either intersects the boundary or is parallel to it. In the latter case $p$ and $q$ 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 $p$ and $q$ 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 ${\mathcal {Z}}\;.$ Distances on ${\mathcal {Z}}$ can be defined using the correspondence with points on ${\bigl \{}(\,1,y\,)\mid y>0{\bigr \}}$ 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 $n$ -space $\,{\mathcal {H}}^{n}\,,$ the maximally symmetric, simply connected, $n$ -dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is $\,{\mathcal {H}}^{2}\,$ 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 $\,{\mathcal {H}}^{n}\,$ of $n$ copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space $\,{\mathcal {H}}_{n}\,,$ which is the domain of Siegel modular forms.

References

Weisstein, Eric W. "Upper Half-Plane". MathWorld.

Complex plane

Affine geometry

Metric geometry

Generalizations

See also

References