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Concept
Riemann sphere
Summary
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point 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 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 infinity. 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. In projective geometry, the sphere can be thought of as the complex projective line , the projective space of all complex lines in . As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics.
The extended complex plane is also called the closed complex plane.
The extended complex numbers consist of the complex numbers together with . The set of extended complex numbers may be written as , and is often denoted by adding some decoration to the letter , such as
The notation has also seen use, but as this notation is also used for the punctured plane , it can lead to ambiguity.
Geometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane).
Addition of complex numbers may be extended by defining, for ,
for any complex number , and multiplication may be defined by
for all nonzero complex numbers , with .
Official source
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