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# Science Fair Project Encyclopedia

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# 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

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

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 $\mathbb{H}_n$ is called the Siegel upper half-space of genus n.