Definitions

1. The open unit disk is commonly used as a model for the hyperbolic plane, by introducing a new metric on it, the Poincaré metric. (Web site)
2. The closed unit disk is a two-dimensional manifold with boundary. (Web site)

Disk Region

1. Each open disk region can be uniformized by a holomorphic map from the open unit disk, which extends continuously to the boundary circle.

Poincaré Metric

1. The unit disk with the Poincaré metric is the unique simply connected oriented 2-dimensional Riemannian manifold with constant curvature -1.

Close Unit Disk

1. So σ(T) lies in the close unit disk of the complex plane. (Web site)

Unit Disk Graph

1. In geometric graph theory, a unit disk graph is the intersection graph of a family of unit circle s in the Euclidean plane.

Solution

1. For example, the solution to the Dirichlet problem for the unit disk in R 2 is given by the Poisson integral formula. (Web site)

Particular

1. In particular, the open unit disk is homeomorphic to the whole plane. (Web site)

Results

1. The same analysis leads to results on the question of whether the canonical divisors in weighted Bergman spaces over the unit disk have extraneous zeros. (Web site)

Continuous

1. The solution u is continuous on the closed unit disk and harmonic on D. (Web site)

Fact

1. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. (Web site)

Uniformly

1. Abstract: We show that the modulus of an inner function can be uniformly approximated in the unit disk by the modulus of an interpolating Blaschke product.

Plane

1. If considered as subspaces of the plane with its standard topology, the open unit disk is an open set and the closed unit disk is a closed set. (Web site)

Theorem

1. Invoking reflexivity and the theorem given above, we can deduce that the open unit disk lies in the residual spectrum of T*. (Web site)
2. Theorem. Let U⊂ C be a simply connected open domain which is not the entire plane of complex number s C, and let D be the unit disk. (Web site)

Identity Map

1. They also obtain bounds for the Belinskii function, which measures the deviation of a quasiconformal automorphism of the unit disk from the identity map.

Functions

1. The author shows how to represent these functions on the closed unit disk using the Cauchy and Poisson integral formulas, thus answering this question. (Web site)

Analytic Functions

1. In this paper we characterize the semigroups of analytic functions in the unit disk which lead to semigroups of operators in the disk algebra.

Unit Sphere

1. As an example we have already seen, consider the operation of taking the unit sphere or unit disk in a vector space with an inner product.

Stereographic Projection

1. In fact, stereographic projection restricted to the projective plane maps onto the unit disk (see Figure 8).

Conformal Map

1. The Riemann mapping theorem states that any open simply connected proper subset of C admits a conformal map on open unit disk in C. (Web site)
2. According to the Riemann mapping theorem, there exists a conformal map from the unit disk to any simply connected planar region (except the whole plane).

Complex Analysis

1. In complex analysis, the Cayley transform is a conformal mapping in which the image of the upper complex half-plane is the unit disk.

Hardy Spaces

1. In complex analysis, the Hardy spaces (or Hardy classes) H p are certain spaces of holomorphic functions on the unit disk or upper half plane.

Schwarz Lemma

1. A variant of the Schwarz lemma can be stated that is invariant under a change of coordinates on the unit disk.
2. Schwarz-Pick theorem A variant of the Schwarz lemma can be stated that is invariant under analytic automorphisms on the unit disk, i.e.

Unit

1. Anal. Appls.,232,1999,272,292,If all the zeros of p lie in the unit disk, then it has a critical point within unit distance of each zero.

Maps

1. Since M(z 1) = 0 and the Möbius transformation is invertible, the composition φ(f(M − 1(z))) maps 0 to 0 and the unit disk is mapped into itself. (Web site)
2. Given U and z 0, we want to construct a function f which maps U to the unit disk and z 0 to 0. (Web site)

Boundary

1. Note that under this projection, antipodal points on the boundary of the unit disk are identified, so the lines shown are actually closed curves.
2. The boundary of the open or closed unit disk is the unit circle. (Web site)

Analytic

1. Analytic and harmonic functions in the unit disk are defined and studied in chapter 3. (Web site)
2. A disk algebra is an algebra of functions which are analytic on the open unit disk in and continuous up to the boundary.
3. The author studies the algebra A of continuous functions on the closed unit disk which are analytic on the open disk in chapter 6. (Web site)

Non-Empty

1. The Riemann mapping theorem states that any non-empty open simply connected subset of C (except for C itself) is conformally equivalent to the unit disk.

Footnotes

1. Footnotes ^ One must use the open unit disk in as the model space instead of because these are not isomorphic, unlike for real manifolds. (Web site)

Quotients

1. All hyperbolic surfaces are quotients of the unit disk.

Holomorphic

1. As a by-product, we also obtain lower bounds for the Taylor coefficients of functions f holomorphic on the unit disk and satisfying. (Web site)

Hyperbolic Space

1. The Poincaré disk model defines a model for hyperbolic space on the unit disk.

Complex Functions

1. We note that the analysis uses theorems of Fatou and Riesz concerning the boundary behavior of complex functions defined on the unit disk. (Web site)

Complex Function

1. A Blaschke product is a complex function which maps the unit disk to itself.

Meromorphic Function

1. A meromorphic function with an infinite number of poles is exemplified by on the punctured disk, where is the open unit disk.

Riemann Surface

1. It is well known that one may study Hardy spaces with the domain a finite bordered Riemann surface rather than the unit disk. (Web site)

Half-Plane

1. On the ordinary complex plane, the Cayley transform carries the upper half-plane to the unit disk, thus bounding it.
2. The unit disk and the upper half-plane are not interchangeable as domains for Hardy spaces.

Complex Plane

1. Let be the open unit disk in the complex plane C. Let be a holomorphic function with.
2. When viewed as a subset of the complex plane (C), the unit disk is often denoted. (Web site)
3. The celebrated Riemann mapping theorem states that any simply connected strict subset of the complex plane is biholomorphic to the unit disk.

Unit Circle

1. For instance, there exists a meromorphic function in the unit disk where every point on the unit circle is a limit point of the set of poles.
2. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle.
3. For example, the closed unit disk is full, while the unit circle is not. (Web site)

Unit Disk

1. In fact, the marked moduli space (Teichmüller space) of the torus is biholomorphic to the upper half-plane or equivalently the open unit disk.
2. In other words every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere.
3. More precisely, a complex manifold has an atlas of charts to the open unit disk[1] in, such that the change of coordinates between charts are holomorphic. (Web site)

Categories

1. Unit Circle
2. Complex Plane
3. Half-Plane
4. Moduli Space
5. Riemann Surface
6. Books about "Unit Disk" in Amazon.com
   

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