# Upper half-plane

In mathematics, the **upper half-plane** **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:

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 **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**.

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 (1) shifts (*x,y*) → (*x* + *c, y*), *c* ∈ ℝ, and (2) 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:**

*Z* can be recognized as the circle of radius 1/2 centered at (1/2, 0), and as the polar plot of

**Proposition:** (0,0), ρ(θ) in *Z*, and (1, tan θ) are collinear points.

In fact, *Z* is the reflection of the line (1,*y*), *y* > 0, in the unit circle. Indeed, the diagonal from (0,0) to (1, tan θ) has squared length so that 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 *Z*. Distances on *Z* can be defined using the correspondence with points on (1,*y*), *y* > 0, 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 **H**^{n}, the maximally symmetric, simply connected, *n*-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is **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 **H**^{n} of *n* copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space **H**_{n}, which is the domain of Siegel modular forms.