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    Riemann Sphere

    Basic definition

    The Riemann sphere is a name given to the complex plane together with a point at infinity, which has the topology of a sphere.

    Detailed definition

    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 {∞^~}, together with two charts. (Here ∞^~ 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 chart is the identity map from the sphere (with infinity removed) to the complex plane. For the point at infinity, the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending infinity to 0 and all other points z to 1/z.

    Related terms

    complex infinity | complex plane | C^* | extended complex plane

    Educational grade level

    college level

    Associated person

    Bernhard Riemann

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