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...

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

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...

\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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of...

{\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)...

coordinate space Complex geometry Complex line Constellation diagram Riemann sphere, which is an extended complex plane s-plane In-phase and quadrature...

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

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

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...

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

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

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

^{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...

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....

and mathematics. 1-sphere, 2-sphere, n-sphere tesseract, polychoron, simplex Pauli matrices Hopf bundle, Riemann sphere Poincaré sphere Reeb foliation Clifford...

Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification...

original setting appearing in Hilbert's twenty-first problem was for the Riemann sphere, where it was about the existence of systems of linear regular differential...