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

corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of...

open upper half-plane. So considered as a Riemann surface, the open unit disk is isomorphic ("biholomorphic", or "conformally equivalent") to the upper half-plane...

whose ⁠ y {\displaystyle y} ⁠ coordinate is greater than zero, the upper half-plane, and a metric tensor (definition of distance) called the Poincaré metric...

extension of f into the upper half-plane. In analogy to the situation for the disk, when u is holomorphic in the upper half-plane, then u is an element...

isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can...

In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by Drinfeld (1976)...

holomorphic on the closed upper half-plane {z ∈ C | Im(z) ≥ 0} such that, for some α > 0, |zα f(z)| is bounded on the closed upper half-plane. Then f ( z ) = 1...

analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such...

surface in question can be taken to be the quotient H/Γ (where H is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified...

In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over...

mathematics, a modular form is a holomorphic function on the complex upper half-plane, H {\displaystyle {\mathcal {H}}} , that roughly satisfies a functional...

\partial D} to D {\displaystyle D} . Let K ⊂ H be a subset of the upper half-plane such that D := H\K is connected and simply connected, and let z ∈ D...

{\displaystyle \operatorname {SL} (2,\mathbb {Z} )} defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic...

half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space...

{\displaystyle H_{+}^{2}(R)} , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula...

{\displaystyle -A} are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations. The name "modular group"...

Look up one half in Wiktionary, the free dictionary. One half is the multiplicative inverse of 2. It is an irreducible fraction with a numerator of 1...

and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary...

rotates the circle. A toy example for a Killing vector field is on the upper half-plane M = R y > 0 2 {\displaystyle M=\mathbb {R} _{y>0}^{2}} equipped with...

hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model...

{\displaystyle H^{p}} are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them...

with the Fourier transform below). For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real...

C_{R}=\{Re^{i\theta }\mid \theta \in [0,\pi ]\}} of positive radius R lying in the upper half-plane, centered at the origin. If the function f is of the form f ( z ) =...

continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis,...

that is, as the quotients of the upper half-plane and a Fuchsian group. For the following, let H be the upper half-plane; let Γ be a Fuchsian group; let...

(t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.} When D {\displaystyle D} is the upper half plane the Loewner equation differs from this by changes of variable and is...

representation theory, and physics. SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through...

common to use 1 {\displaystyle 1} and τ {\displaystyle \tau } in the upper half-plane H := { z ∈ C : Im ⁡ ( z ) > 0 } {\displaystyle \mathbb {H} :=\{z\in...

curvature. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane, a geometry of constant curvature whose lines...