Riemann surface

Riemann surface
a one-dimensional complex manifold; a generalization of the complex plane

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  • Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

  • riemann surface — noun Usage: usually capitalized R Etymology: after G.F.B. Riemann : a multilayered surface in the theory of complex functions on which a multivalued complex function can be treated as a single valued function of a complex variable * * * Math. a… …   Useful english dictionary

  • Riemann surface — Math. a geometric representation of a function of a complex variable in which a multiple valued function is depicted as a single valued function on several planes, the planes being connected at some of the points at which the function takes on… …   Universalium

  • Compact Riemann surface — In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact (those that are closed manifolds) and the open (the rest, which from the… …   Wikipedia

  • Riemann sphere — 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 (or extended complex plane), named after the 19th… …   Wikipedia

  • Surface — This article discusses surfaces from the point of view of topology. For other uses, see Differential geometry of surfaces, algebraic surface, and Surface (disambiguation). An open surface with X , Y , and Z contours shown. In mathematics,… …   Wikipedia

  • Riemann–Roch theorem — In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates… …   Wikipedia

  • Riemann hypothesis — The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011 …   Wikipedia

  • Riemann, Bernhard — ▪ German mathematician in full  Georg Friedrich Bernhard Riemann  born September 17, 1826, Breselenz, Hanover [Germany] died July 20, 1866, Selasca, Italy  German mathematician whose profound and novel approaches to the study of geometry laid the …   Universalium

  • Riemann mapping theorem — In complex analysis, the Riemann mapping theorem states that if U is a simply connected open subset of the complex number plane Bbb C which is not all of Bbb C, then there exists a biholomorphic (bijective and holomorphic) mapping f, from U, onto …   Wikipedia

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