Upper half-plane#Complex plane

{{Short description|Complex numbers with non-negative imaginary part}}

In mathematics, the upper half-plane, {{tmath|\mathcal H,}} is the set of points {{tmath|(x,y)}} in the Cartesian plane with {{tmath|y > 0.}} The lower half-plane is the set of points {{tmath|(x,y)}} with {{tmath|y < 0}} instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants.

Affine geometry

The affine transformations of the upper half-plane include

shifts $(x,y)\mapsto (x+c,y)$ , $c\in\mathbb{R}$ , and
dilations $(x,y)\mapsto (\lambda x,\lambda y)$ , $\lambda > 0.$

Proposition: Let {{tmath|A}} and {{tmath|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 {{tmath|A}} to {{tmath|(0,0).}} Then take $\lambda=(\text{diameter of}\ B)/(\text{diameter of}\ A)$

and dilate. Then shift {{tmath|(0,0)}} to the center of {{tmath|B.}}

Inversive geometry

Definition: $\mathcal{Z} := \left\{\left( \cos^2\theta,\tfrac12 \sin 2\theta \right) \mid 0 < \theta < \pi \right\}$ .

{{tmath|\mathcal Z}} can be recognized as the circle of radius {{tmath|\tfrac12}} centered at {{tmath|\bigl(\tfrac12,0\bigr),}} and as the polar plot of $\rho(\theta) = \cos \theta.$

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

In fact, $\mathcal{Z}$ is the inversion of the line $\bigl\{(1, y) \mid y > 0 \bigr\}$ in the unit circle. Indeed, the diagonal from {{tmath|(0,0)}} to {{tmath|(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 {{tmath|p}} and {{tmath|q}} in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from {{tmath|p}} to {{tmath|q}} either intersects the boundary or is parallel to it. In the latter case {{tmath|p}} and {{tmath|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 {{tmath|p}} and {{tmath|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

{{tmath|\mathcal Z.}} Distances on {{tmath|\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.

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:

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

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 {{tmath|y < 0}} is equally good, but less used by convention. The open unit disk {{tmath|\mathcal D}} (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to {{tmath|\mathcal H}} (see "Poincaré metric"), meaning that it is usually possible to pass between

{{tmath|\mathcal H}} and {{tmath|\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.

Generalizations

One natural generalization in differential geometry is hyperbolic $n$ -space {{tmath|\mathcal H^n,}} the maximally symmetric, simply connected, {{tmath|n}}-dimensional Riemannian manifold with constant sectional curvature $-1$ . In this terminology, the upper half-plane is

{{tmath|\mathcal H^2}} since it has real dimension {{tmath|2.}}

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