Search results
Results From The WOW.Com Content Network
According to this weaker definition, a conformal map need not be biholomorphic, even though it is locally biholomorphic, for example, by the inverse function theorem. For example, if f: U → U is defined by f(z) = z 2 with U = C–{0}, then f is conformal on U, since its derivative f’(z) = 2z ≠ 0, but it is not biholomorphic, since it is 2-1.
In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic mapping (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from onto the open unit disk
Since the transition maps between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not just orientable: a biholomorphic map to (a subset of) C n gives an orientation, as biholomorphic maps are orientation-preserving).
Biholomorphic map; Carathéodory's theorem – A conformal map extends continuously to the boundary; Penrose diagram; Schwarz–Christoffel mapping – a conformal transformation of the upper half-plane onto the interior of a simple polygon; Special linear group – transformations that preserve volume (as opposed to angles) and orientation
The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible conformal map (i.e. biholomorphic map) from the Riemann sphere to itself.
The search engine that helps you find exactly what you're looking for. Find the most relevant information, video, images, and answers from all across the Web.
This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps D → D. By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the Poincaré disk model , and this group is the group of all orientation-preserving isometries of H 2 in this ...
An interactive map showing how opioid abuse rates outpace treatment capacity 2 to 1. 350 Miles For Treatment.