The term arises from a common visualization of the complex number x + iy as the point (x, y) in the plane endowed with Cartesian coordinates. When the Y-axis is oriented vertically, the "upper half-plane" corresponds to the region above the X-axis and thus complex numbers for which 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 (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to H (see "Poincaré metric"), 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 affine transformations of the upper half-plane include (1) shifts (x,y) -> (x + c, y), c ? R, and (2) dilations (x,y) -> (λ x, λ y), λ > 0.
Proposition: Let A and B be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B.
Z can be recognized as the circle of radius 1/2 centered at (1/2, 0), and as the polar plot of
Proposition: (0,0), ρ(θ) in Z, and (1, tan θ) are collinear points.
In fact, Z is the reflection of the line (1,y), y > 0, in the unit circle. Indeed, the diagonal from (0,0) to (1, tan θ) has squared length so that is the reciprocal of that length.
The distance between any two points p and q in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from p to q either intersects the boundary or is parallel to it. In the latter case p and q lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation. In the former case p and q lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to Z. Distances on Z can be defined using the correspondence with points on (1,y), y > 0, and logarithmic measure on this ray. In consequence, the upper half-plane becomes a metric space. The generic name of this metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model.
One natural generalization in differential geometry is hyperbolic n-space Hn, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature -1. In this terminology, the upper half-plane is H2 since it has real dimension 2.
In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product Hn of n copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space Hn, which is the domain of Siegel modular forms.