Science Fair Project Encyclopedia
Upper half plane
- x + iy
- 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.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details