In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the...

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...

\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...

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...

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...

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...

the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they...

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...

transformations are the conformal transformations of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of SL(2, C)...

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...

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...

representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open...

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...

The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension...

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...

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...

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....

^{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...

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...

Parallelizable 2-sphere Commonly simply called a sphere. For its complex structure, see Riemann sphere. Homeomorphic to the complex projective line 3-sphere Parallelizable...

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...

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...

a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those...

naturally extended to a function whose domain and range are the whole Riemann sphere (complex projective line). A complex rational function with degree one...

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...

contour, meaning a path that begins and ends at the point ∞ on the Riemann sphere, whose unit tangent vector converges to −1 at the start of the path...

Georg Friedrich Bernhard Riemann (/ˈriːmɑːn/; German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] ; 17 September 1826 – 20 July 1866) was a German mathematician...

{\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)...

two-dimensional spherical surface, when given a complex metric, becomes a Riemann sphere of one complex dimension. The dimension of an algebraic variety may...

sphere Hoberman sphere Homology sphere Homotopy groups of spheres Homotopy sphere Lenart Sphere Napkin ring problem Orb (optics) Pseudosphere Riemann...