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

<math>\mathbb{H}_n=\{F\in M(n,C) \; s.t. F=F^T \;\textrm{and}\; \Im F >0 \}<math>

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 <math>\mathbb{H}_n<math> 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