Upper half plane
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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.