In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the...
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different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include graphs of multivalued...
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transformed into a perfect circle by some conformal map. A map of the Riemann sphere onto itself is conformal if and only if it is a Möbius transformation...
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\mathbf {P} ^{1}.} This is the Bloch sphere, which can be mapped to the Riemann sphere. The Bloch sphere is a unit 2-sphere, with antipodal points corresponding...
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a point at infinity is called the Riemann sphere. If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros...
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Rhumb line (section On the Riemann sphere)
the Earth can be understood mathematically as a Riemann sphere, that is, as a projection of the sphere to the complex plane. In this case, loxodromes can...
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complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere), and then the extra point gives a 3-dimensional version of a...
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Möbius transformation (category Riemann surfaces)
the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they...
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transformations are the conformal transformations of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of SL(2, C)...
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Meromorphic function (section On Riemann surfaces)
a meromorphic function can be defined for every Riemann surface. When D is the entire Riemann sphere, the field of meromorphic functions is simply the...
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that is topologically a sphere. Hence the complex projective line is also known as the Riemann sphere (or sometimes the Gauss sphere). It is in constant use...
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Uniformization theorem (category Riemann surfaces)
connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem...
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Kleinian group (redirect from Sphere at infinity)
representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open...
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The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension...
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notation. He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with monodromy group P S L ( 2 , 11 ) {\displaystyle PSL(2...
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Division by zero (section Riemann sphere)
consequence, the set of extended complex numbers is often called the Riemann sphere. The set is usually denoted by the symbol for the complex numbers decorated...
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^{1}} be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann sphere. Let V = T...
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Pn(C) or CPn. When n = 1, the complex projective space CP1 is the Riemann sphere, and when n = 2, CP2 is the complex projective plane (see there for...
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concurrent lines on the projective plane and a quadruple of points on the Riemann sphere. In the Cayley–Klein model of hyperbolic geometry, the distance between...
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Parallelizable 2-sphere Commonly simply called a sphere. For its complex structure, see Riemann sphere. Homeomorphic to the complex projective line 3-sphere Parallelizable...
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any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other....
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For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity...
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The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...
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Inversive geometry (redirect from Sphere inversion)
model for the Möbius plane that comes from the Euclidean plane is the Riemann sphere. The cross-ratio between 4 points x , y , z , w {\displaystyle x,y,z...
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a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those...
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and mathematics. 1-sphere, 2-sphere, n-sphere tesseract, polychoron, simplex Pauli matrices Hopf bundle, Riemann sphere Poincaré sphere Reeb foliation Clifford...
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domain and range are the whole Riemann sphere, i.e., a rational mapping. Iteration of rational functions on the Riemann sphere forms a discrete dynamical...
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Millennium Prize Problems (section Riemann hypothesis)
conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture...
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{\displaystyle \mathbb {C} } (but is sometimes called the Riemann sphere, as it is a model of the sphere, two-dimensional with respect to real-number coordinates)...
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Fundamental polygon (category Riemann surfaces)
universal covering surface given by exactly one of the following: the Riemann sphere, the complex plane, the unit disk D or equivalently the upper half-plane...
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