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

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = { ⟨ x , y ⟩ ∣ y > 0 ; x , y ∈ R } {\displaystyle...

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...

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...

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...

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

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...

mathematics, a modular form is a (complex) analytic function on the upper half-plane, H {\displaystyle \,{\mathcal {H}}\,} , that satisfies: a kind of functional...

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...

Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them...

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...

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)...

function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic...

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...

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...

matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular...

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...

(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...

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

{\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...

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,...

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...

is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs...

Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to...

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

representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such...

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...

Euclidean plane may be taken to be a plane with the Cartesian coordinate system and the x-axis is taken as line B and the half plane is the upper half (y >...

the following: the Riemann sphere, the complex plane, the unit disk D or equivalently the upper half-plane H. In the first case of genus zero, the surface...