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Riemann sphere

Riemann sphere noun: the complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.

Noun

Riemann sphere (plural Riemann spheres)

  1. The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.
    • We use #92;hat#123;#92;Complex#125;#92;times (resp. #92;Complex) to denote the subset #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne 0#92;#125; (resp. #92;#123;z#92;in#92;hat#92;Complex#58;z#92;ne#92;infty#92;#125;) in the Riemann sphere #92;hat#92;Complex.2002, Wenhua Zhao, “Some Generalizations of Genus Zero Two-Dimensional Conformal Field Theory”, in Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, Brian Parshall, editors, Recent...
    Synonyms:
    extended complex numbers, closed complex plane, extended complex plane
  2. The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.
    • 1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,
      Every circle γ on the Riemann sphere Σ which does not go through a given point P^*∈Σ divides Σ into two parts, such that one part contains P^* and the other does not.
    • In order to visualize the point at infinity, we consider the Riemann sphere that has radius 1#47;2 and is tangent to the complex plane at the origin (see Figure 1.8). We call the point of contact the south pole (denoted by S) and the point diametrically opposite S the north pole (denoted by N). Let...2002, Yue Kuen Kwok, Applied Complex Variables for Scientists and Engineers, Cambridge University Press, page 27

Riemann sphere meaning in Tamil

Riemann sphere in Tamil

ரைமான் கோளம்

Examples of Riemann sphere in a sentence

  • In order to visualize the point at infinity, we consider the Riemann sphere that has radius 1#47;2 and is tangent to the complex plane at the origin (see Figure 1.8).2002, Yue Kuen Kwok, Applied Complex Variables for Scientists and Engineers, Cambridge University Press, page 27:
  • We find that many mathematicians, even those who specialize in complex analysis and conformal geometry, are not familiar with the inversive distance between pairs of circles in the Riemann sphere.2003, Philip L. Bowers, Monica K. Hurdal, “Planar Conformal Mappings of Piecewise Flat Surfaces”, in Hans-Christian Hege, Konrad Polthier...

Synonyms of Riemann sphere

  • extended complex numbers
  • closed complex plane
  • extended complex plane

Forms of Riemann sphere

pluralRiemann spheres

Origin of Riemann sphere

Named after German mathematician Bernhard Riemann.

Questions about Riemann sphere

What does Riemann sphere mean?
Riemann sphere means the complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.
What is Riemann sphere in Tamil?
Riemann sphere in Tamil is ரைமான் கோளம். See Riemann sphere meaning in Tamil.
How do you use Riemann sphere in a sentence?
In order to visualize the point at infinity, we consider the Riemann sphere that has radius 1#47;2 and is tangent to the complex plane at the origin (see Figure 1.8).
What part of speech is Riemann sphere?
Riemann sphere is a noun.
What is the plural of Riemann sphere?
The plural of Riemann sphere is Riemann spheres.

Words near Riemann sphere in the dictionary

  1. Riemann hypothesis
  2. Riemann integral
  3. Riemann integrals
  4. Riemann space
  5. Riemann spaces
  6. Riemann sphere
  7. Riemann spheres
  8. Riemann sum
  9. Riemann surface
  10. Riemann surfaces
  11. Riemann zeta function

Source and license

Adapted from Wiktionary: Riemann sphere (page history and authors); riemann sphere (Tamil Wiktionary) (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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Cite this entry: “Riemann sphere.” Word.ws, https://word.ws/riemann_sphere. Based on Wiktionary, CC BY-SA 4.0.