Upper half-plane

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In mathematics, the upper half-plane, ${\displaystyle \,{\mathcal {H}}\,,}$ is the set of points (x, y) in the Cartesian plane with y > 0.

Complex plane[]

Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part:

${\displaystyle {\mathcal {H}}\equiv \{x+iy\mid y>0;x,y\in \mathbb {R} \}~.}$

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 ${\displaystyle \,{\mathcal {D}}\,}$ (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to ${\displaystyle \,{\mathcal {H}}\,}$ (see "Poincaré metric"), meaning that it is usually possible to pass between ${\displaystyle \,{\mathcal {H}}\,}$ and ${\displaystyle \,{\mathcal {D}}\;.}$

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 uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature.

The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.

Affine geometry[]

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.

Proof: First shift the center of A to (0,0). Then take λ = (diameter of B)/(diameter of A) and dilate. Then shift (0,0) to the center of B.

Definition: ${\displaystyle \;{\mathcal {Z}}\equiv {\bigl \{}\,\left(\,\cos ^{2}\theta \,,\;{\tfrac {1}{2}}\sin 2\theta \,\right)\;|\;0<\theta <\pi \,{\bigr \}}~.}$

${\displaystyle \,{\mathcal {Z}}\,}$ can be recognized as the circle of radius ½ centered at (½, 0), and as the polar plot of ${\displaystyle \rho (\theta )=\cos \theta ~.}$

Proposition: (0,0), ${\displaystyle \,\rho (\theta )\,}$ in ${\displaystyle \,{\mathcal {Z}}\,,}$ and ${\displaystyle \,(\,1,\tan \theta \,)\,}$ are collinear points.

In fact, ${\displaystyle \,{\mathcal {Z}}\,}$ is the reflection of the line ${\displaystyle {\bigl \{}\,(\,1,y\,)\,|\,y>0\,{\bigr \}}\,}$ in the unit circle. Indeed, the diagonal from (0,0) to ${\displaystyle \,(\,1,\tan \theta \,)\,}$ has squared length ${\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta \,,}$ so that ${\displaystyle \,\rho (\theta )=\cos \theta \,}$ is the reciprocal of that length.

Metric geometry[]

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 ${\displaystyle \,{\mathcal {Z}}\;.}$ Distances on ${\displaystyle \,{\mathcal {Z}}\,}$ can be defined using the correspondence with points on ${\displaystyle {\bigl \{}\,(\,1,y\,)\,|\,y>0\,{\bigr \}}\,}$ 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.

Generalizations[]

One natural generalization in differential geometry is hyperbolic n-space ${\displaystyle \,{\mathcal {H}}^{n}\,,}$ the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is ${\displaystyle \,{\mathcal {H}}^{2}\,}$ 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 ${\displaystyle \,{\mathcal {H}}^{n}\,}$ of n copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space ${\displaystyle \,{\mathcal {H}}_{n}\,,}$ which is the domain of Siegel modular forms.

See also[]

• Cusp neighborhood
• Extended complex upper-half plane
• Fuchsian group
• Fundamental domain
• Half-space
• Kleinian group
• Modular group
• Riemann surface
• Schwarz–Ahlfors–Pick theorem
• Moduli stack of elliptic curves

References[]

• Weisstein, Eric W. "Upper Half-Plane". MathWorld.