PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (195)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
Runge's theorem (Theorem)

Let $ K$ be a compact subset of $ \mathbb{C}$, and let $ E$ be a subset of $ \mathbb{C}_\infty=\mathbb{C}\cup\{\infty\}$ (the extended complex plane) which intersects every connected component of $ \mathbb{C}_\infty-K$. If $ f$ is an analytic function in an open set containing $ K$, given $ \varepsilon>0$, there is a rational function $ R(z)$ whose only poles are in $ E$, such that $ \vert f(z)-R(z)\vert<\varepsilon$ for all $ z\in K$.



"Runge's theorem" is owned by Koro.
(view preamble)

View style:

See Also: Mergelyan's theorem
Log in to rate this entry.
(view current ratings)

Cross-references: poles, rational function, open set, analytic function, connected component, intersects, extended complex plane, subset, compact subset
There are 2 references to this entry.

This is version 3 of Runge's theorem, born on 2002-12-11, modified 2004-06-07.
Object id is 3731, canonical name is RungesTheorem.
Accessed 2770 times total.

Classification:
AMS MSC: 30E10 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Approximation in the complex domain)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)