The Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point corresponding to 1/0. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value 1/0. With the Riemann model, the point 1/0 is near to very large numbers, just as the point "0" 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 1/0 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 1/0. 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.
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