Upper half-plane


Upper half-plane

In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part y:

\mathbb{H} = \{x + iy \;| y > 0; x, y \in \mathbb{R} \}.

The term is associated with a common visualization of complex numbers with points in the plane endowed with Cartesian coordinates, with the Y-axis pointing upwards: the "upper half-plane" corresponds to the half-plane above the X-axis.

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 (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.

Generalizations

One natural generalization in differential geometry is hyperbolic n-space Hn, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is H2 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 Hn of n copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space Hn, which is the domain of Siegel modular forms.

See also

References


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Lower half-plane — In mathematics, the lower half plane H is the set of complex numbers :mathbb{H} = {x + iy ;| y < 0; x, y in mathbb{R} } with positive imaginary part y . Other names are hyperbolic plane, Poincaré plane and Lobachevsky plane, particularly in texts …   Wikipedia

  • Poincaré half-plane model — Stellated regular heptagonal tiling of the model.In non Euclidean geometry, the Poincaré half plane model is the upper half plane, together with a metric, the Poincaré metric, that makes it a model of two dimensional hyperbolic geometry.It is… …   Wikipedia

  • Siegel upper half-space — In mathematics, the Siegel upper half space of degree n is the set of n times; n symmetric matrices over the complex number field whose imaginary part is positive definite. It is named in honor of Carl Ludwig Siegel.In the case n = 1, the Siegel… …   Wikipedia

  • Half-space — For other uses, see Half space (disambiguation). In geometry, a half space is either of the two parts into which a plane divides the three dimensional euclidean space. More generally, a half space is either of the two parts into which a… …   Wikipedia

  • Half-period ratio — In mathematics, the half period ratio tau; of an elliptic function j is the ratio : au = frac{omega 2}{omega 1}of the two half periods omega;1 and omega;2 of j , where j is defined in such a way that:Im( au) > 0is in the upper half plane.Note… …   Wikipedia

  • Plane (Dungeons & Dragons) — Ethereal plane redirects here. For the mystical concept, see Etheric plane. The planes of the Dungeons Dragons roleplaying game constitutes the multiverse in which the game takes place. In the earliest versions of Dungeons Dragons, the concept of …   Wikipedia

  • Moore plane — In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii s tangent disk topology) is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) which is not normal.… …   Wikipedia

  • Lobachevski plane — #redirect Upper half plane …   Wikipedia

  • Complex plane — Geometric representation of z and its conjugate in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z. In mathematics …   Wikipedia

  • Ground plane — In electrical engineering, a ground plane is an electrically conductive surface. Radio antenna theory In telecommunication, a ground plane structure or relationship exists between the antenna and another object, where the only structure of the… …   Wikipedia


We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.