# Upper half-plane

In mathematics, the **upper half-plane** **H** is 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.

## 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.

## See also

- Cusp neighborhood
- Extended complex upper-half plane
- Fuchsian group
- Fundamental domain
- Half space
- Hyperbolic geometry, especially the Poincaré half-plane model
- Kleinian group
- Modular group
- Riemann surface
- Schwarz-Ahlfors-Pick theorem