Upper half-plane

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

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

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 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 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 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

References

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