Complex 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.

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