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Riemann Sphere

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers C^*=C union {infty^~}, together with two coordinate charts. (Here infty^~ denotes complex infinity.) The notation C^^ is also used (Krantz 1999, p. 82; Lorentzen and Waadeland 2008, p. 3).

For all points in the complex plane, the coordinate chart is the identity map from the sphere (with infinity removed) to the complex plane. For the point at infinity, the coordinate chart neighborhood is the sphere (with the origin removed), and the coordinate chart is given by sending infinity to 0 and all other points z to 1/z.

See also

C*, Complex Infinity, Complex Plane, Extended Complex Plane, Möbius Transformation, Residue at Infinity Explore this topic in the MathWorld classroom

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References

Anderson, J. W. "The Riemann Sphere C^_." §1.2 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 7-16, 1999.Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, p. 4, 1996.Krantz, S. G. "The Riemann Sphere." §6.3.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 83-84, 1999.Lorentzen, L. and Waadeland, H. Continued Fractions, 2nd ed., Vol. 1: Convergence Theory. Amsterdam, Netherlands/Paris, France: Atlantis Press/World Scientific, 2008.

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Riemann Sphere

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Weisstein, Eric W. "Riemann Sphere." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannSphere.html

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