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