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Riemann sphere (Definition)

The Riemann Sphere, denoted $ \hat{\mathbb{C}}$ is the one-point compactification of the complex plane $ \mathbb{C}$, obtained by identifying the limits of all infinitely extending rays from the origin as one single “point at infinity.” Heuristically, $ \hat{\mathbb{C}}$ can be viewed as a 2-sphere with the top point corresponding to the point at infinity, and the bottom point corresponding the origin. An atlas for the Riemann sphere is given by two charts:

$\displaystyle \hat{\mathbb{C}}\backslash\{\infty\}\rightarrow\mathbb{C}:z\mapsto z$    

and
$\displaystyle \hat{\mathbb{C}}\backslash\{0\}\rightarrow\mathbb{C}:z\mapsto \frac{1}{z}$    

Any rational function on $ \hat{\mathbb{C}}$ has a unique smooth extension to a map $ \hat{p}:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}$.



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See Also: complex plane, complex, when all singularities are poles, closed complex plane

Keywords:  compactification

Pronunciation (guide):
 Riemann sphere: /ree-mawn/

Attachments:
zeros and poles of rational function (Topic) by pahio
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Cross-references: map, extension, smooth, rational function, charts, atlas, infinity, point, origin, rays, limits, complex plane, one-point compactification
There are 14 references to this entry.

This is version 6 of Riemann sphere, born on 2003-07-17, modified 2007-11-21.
Object id is 4469, canonical name is RiemannSphere.
Accessed 7833 times total.

Classification:
AMS MSC32C15 (Several complex variables and analytic spaces :: Analytic spaces :: Complex spaces)

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