PlanetMath: Riemann sphere

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (37)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

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 $\hat{p}:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}$ .

"Riemann sphere" is owned by mathcam.

(view preamble)

Keywords:

compactification

Pronunciation (guide):

Riemann sphere: /ree-mawn/

Attachments:: zeros and poles of rational function (Topic) by pahio

Log in to rate this entry.
(view current ratings)

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 MSC:

32C15 (Several complex variables and analytic spaces :: Analytic spaces :: Complex spaces)

Pending Errata and Addenda

None.[ View all 5 ]

Discussion

forum policy

No messages.

Interact