Model of the extended complex plane plus a point at infinity
The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below).
In mathematics, the Riemann sphere, named after Bernhard Riemann,[1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers.
The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.
In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line, the projective space of all complex lines in . As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics.
^B. Riemann: Theorie der Abel'sche Funktionen, J. Math. (Crelle) 1857; Werke 88-144. The name is due to Neumann C :Vorlesungen über Riemanns Theorie der Abelsche Integrale, Leipzig 1865 (Teubner)
In mathematics, the Riemannsphere, 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...
a point at infinity is called the Riemannsphere. If f is a function that is meromorphic on the whole Riemannsphere, then it has a finite number of zeros...
\mathbf {P} ^{1}.} This is the Bloch sphere, which can be mapped to the Riemannsphere. The Bloch sphere is a unit 2-sphere, with antipodal points corresponding...
a meromorphic function can be defined for every Riemann surface. When D is the entire Riemannsphere, the field of meromorphic functions is simply the...
complex numbers instead, the corresponding projective "line" is a sphere (the Riemannsphere), and then the extra point gives a 3-dimensional version of a...
connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemannsphere. The theorem...
transformed into a perfect circle by some conformal map. A map of the Riemannsphere onto itself is conformal if and only if it is a Möbius transformation...
that is topologically a sphere. Hence the complex projective line is also known as the Riemannsphere (or sometimes the Gauss sphere). It is in constant use...
the Earth can be understood mathematically as a Riemannsphere, that is, as a projection of the sphere to the complex plane. In this case, loxodromes can...
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...
transformations are the conformal transformations of the Riemannsphere (or celestial sphere). Then conjugating with an arbitrary element of SL(2, C)...
consequence, the set of extended complex numbers is often called the Riemannsphere. The set is usually denoted by the symbol for the complex numbers decorated...
a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemannsphere. Arithmetic operations similar to those...
sphere Hoberman sphere Homology sphere Homotopy groups of spheres Homotopy sphere Lenart Sphere Napkin ring problem Orb (optics) Pseudosphere Riemann...
For example, it is easy to prove that the analytic functions from the Riemannsphere to itself are either the rational functions or the identically infinity...
any two simply connected open subsets of the Riemannsphere which both lack at least two points of the sphere can be conformally mapped into each other....
field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...
^{1}} be the Riemannsphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemannsphere. Let V = T...
Julia set Jc, so it is a convex combination of two points in Jc. In the Riemannsphere polynomial has 2d-2 critical points. Here zero and infinity are critical...
representation as orientation-preserving conformal transformations of the Riemannsphere, and as orientation-preserving conformal transformations of the open...
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