Upper half plane
In mathematics, the upper half plane H is the set of complex numbers
- x + iy
with real number x and y, such that the imaginary part
- 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 is equivalent by a conformal mapping, 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 multi-dimensional analog of the upper half-plane is the Siegel upper half-space. Let
be set of symmetric square matrices whose imaginary part is positive definite; that is the set of square matrices whose imaginary parts have positive eigenvalues. The set 
is called the Siegel upper half-space of genus n.
See also
- Cusp neighborhood
- Fuchsian group
- Fundamental domain
- Hyperbolic geometry
- Kleinian group
- Modular group
- Poincaré metric
- Riemann surface
- Schwarz-Ahlfors-Pick theorem