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In particular, the open unit disk is homeomorphic to the whole plane. There is however no conformal bijective map between the open unit disk and the plane. Considered as a Riemann surface, the open unit disk is therefore different from the complex plane. There are conformal bijective maps between the open unit disk and the open upper half-plane ...
While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.
The Riemann mapping theorem can be generalized to the context of Riemann surfaces: If is a non-empty simply-connected open subset of a Riemann surface, then is biholomorphic to one of the following: the Riemann sphere, the complex plane, or the unit disk.
The simply connected 1-dimensional complex manifolds are isomorphic to either: . Δ, the unit disk in C; C, the complex plane; Ĉ, the Riemann sphere; Note that there are inclusions between these as Δ ⊆ C ⊆ Ĉ, but that there are no non-constant holomorphic maps in the other direction, by Liouville's theorem.
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
Replacing simplices with disks of various dimensions results in a related construction called cellular homology. There are also other ways of computing these homology groups, for example via Morse homology , or by taking the output of the Universal Coefficient Theorem when applied to a cohomology theory such as Čech cohomology or (in the case ...
The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.
In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).