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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
Cusp neighborhood
Extended complex upper-half plane
Fuchsian group
Fundamental domain
Hyperbolic geometry
Kleinian group
Modular group
Riemann surface
Schwarz-Ahlfors-Pick theorem
References
Weisstein, Eric W., "Upper Half-Plane", MathWorld.
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