Upper half-plane

The affine transformations of the upper half-plane include

1. shifts ${\displaystyle (x,y)\mapsto (x+c,y)}$, ${\displaystyle c\in \mathbb {R} }$, and
2. dilations ${\displaystyle (x,y)\mapsto (\lambda x,\lambda y)}$, ${\displaystyle \lambda >0.}$

Proposition: Let ⁠${\displaystyle A}$⁠ and ⁠${\displaystyle B}$⁠ be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes ${\displaystyle A}$ to ${\displaystyle B}$.

Proof: First shift the center of ⁠${\displaystyle A}$⁠ to ⁠${\displaystyle (0,0).}$⁠ Then take ${\displaystyle \lambda =({\text{diameter of}}\ B)/({\text{diameter of}}\ A)}$

and dilate. Then shift ⁠${\displaystyle (0,0)}$⁠ to the center of ⁠${\displaystyle B.}$⁠

Definition: ${\displaystyle {\mathcal {Z}}:=\left\{\left(\cos ^{2}\theta ,{\tfrac {1}{2}}\sin 2\theta \right)\mid 0<\theta <\pi \right\}}$.

⁠${\displaystyle {\mathcal {Z}}}$⁠ can be recognized as the circle of radius ⁠${\displaystyle {\tfrac {1}{2}}}$⁠ centered at ⁠${\displaystyle {\bigl (}{\tfrac {1}{2}},0{\bigr )},}$⁠ and as the polar plot of ${\displaystyle \rho (\theta )=\cos \theta .}$

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

In fact, ${\displaystyle {\mathcal {Z}}}$ is the inversion of the line ${\displaystyle {\bigl \{}(1,y)\mid y>0{\bigr \}}}$ in the unit circle. Indeed, the diagonal from ⁠${\displaystyle (0,0)}$⁠ 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.

The distance between any two points ⁠${\displaystyle p}$⁠ and ⁠${\displaystyle q}$⁠ in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from ⁠${\displaystyle p}$⁠ to ⁠${\displaystyle q}$⁠ either intersects the boundary or is parallel to it. In the latter case ⁠${\displaystyle p}$⁠ and ⁠${\displaystyle q}$⁠ 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 ⁠${\displaystyle p}$⁠ and ⁠${\displaystyle q}$⁠ 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 ${\displaystyle {\bigl \{}(1,y)\mid y>0{\bigr \}}}$ 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.

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:

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

The term arises from a common visualization of the complex number ${\displaystyle x+iy}$ as the point ${\displaystyle (x,y)}$ in the plane endowed with Cartesian coordinates. When the ${\displaystyle y}$ axis is oriented vertically, the "upper half-plane" corresponds to the region above the ${\displaystyle x}$ axis and thus complex numbers for which ${\displaystyle y>0}$.

It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by ⁠${\displaystyle y<0}$⁠ 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.

One natural generalization in differential geometry is hyperbolic ${\displaystyle n}$-space ⁠${\displaystyle {\mathcal {H}}^{n},}$⁠ the maximally symmetric, simply connected, ⁠${\displaystyle n}$⁠-dimensional Riemannian manifold with constant sectional curvature ${\displaystyle -1}$. In this terminology, the upper half-plane is ⁠${\displaystyle {\mathcal {H}}^{2}}$⁠ since it has real dimension ⁠${\displaystyle 2.}$⁠

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 ⁠${\displaystyle n}$⁠ 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.

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