the upper half-plane, H , {\displaystyle {\mathcal {H}},} is the set of points ( x , y ) {\displaystyle (x,y)} in the Cartesian plane with ...
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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...
21 KB (3,560 words) - 20:56, 18 May 2023
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...
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Modular curve (redirect from Extended complex upper-half plane)
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...
15 KB (2,016 words) - 12:06, 8 November 2023
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...
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Poisson kernel (section On 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...
9 KB (1,481 words) - 16:09, 28 May 2024
{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...
9 KB (1,021 words) - 15:31, 28 July 2024
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...
11 KB (1,557 words) - 00:31, 27 May 2023
mathematics, a modular form is a (complex) analytic function on the upper half-plane, H {\displaystyle \,{\mathcal {H}}\,} , that satisfies: a kind of functional...
31 KB (4,554 words) - 18:58, 30 July 2024
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...
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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...
30 KB (4,345 words) - 04:25, 11 November 2023
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...
8 KB (913 words) - 13:06, 26 December 2023
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)...
1 KB (177 words) - 17:15, 24 November 2014
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...
27 KB (4,703 words) - 11:45, 7 August 2024
Schwarz integral formula (section Upper half-plane)
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...
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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...
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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...
25 KB (3,317 words) - 11:59, 8 November 2023
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...
23 KB (3,018 words) - 13:18, 9 July 2024
(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...
22 KB (2,994 words) - 13:52, 11 August 2024
with the Fourier transform below). For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real...
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{\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...
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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,...
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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...
4 KB (447 words) - 20:26, 21 August 2021
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...
17 KB (2,943 words) - 21:52, 27 July 2024
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...
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representation theory, and physics. SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through...
21 KB (2,988 words) - 18:22, 23 July 2024
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...
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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...
27 KB (4,720 words) - 21:59, 3 August 2024
Hyperbolic geometry (redirect from Hyperbolic plane)
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 >...
56 KB (7,019 words) - 21:44, 8 August 2024
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...
43 KB (5,995 words) - 15:05, 31 March 2024