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- Runge's_theorem abstract "In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following: Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex number from every bounded connected component of C\K then there exists a sequence of rational functions which converges uniformly to f on K and such that all the poles of the functions are in A. Note that not every complex number in A needs to be a pole of every rational function of the sequence . We merely know that for all members of that do have poles, those poles lie in A.One aspect that makes this theorem so powerful is that one can choose the set A arbitrarily. In other words, one can choose any complex numbers from the bounded connected components of C\K and the theorem guarantees the existence of a sequence of rational functions with poles only amongst those chosen numbers.For the special case in which C\K is a connected set (or equivalently that K is simply-connected), the set A in the theorem will clearly be empty. Since rational functions with no poles are simply polynomials, we get the following corollary: If K is a compact subset of C such that C\K is a connected set, and f is a holomorphic function on K, then there exists a sequence of polynomials that approaches f uniformly on K.Runge's theorem generalises as follows: if one takes A to be a subset of the Riemann sphere C∪{∞} and requires that A intersect also the unbounded connected component of K (which now contains ∞). That is, in the formulation given above, the rational functions may turn out to have a pole at infinity, while in the more general formulation the pole can be chosen instead anywhere in the unbounded connected component of K.".
- Runge's_theorem thumbnail Runge_theorem.svg?width=300.
- Runge's_theorem wikiPageID "3199591".
- Runge's_theorem wikiPageRevisionID "561421124".
- Runge's_theorem hasPhotoCollection Runge's_theorem.
- Runge's_theorem id "p/r082830".
- Runge's_theorem title "Runge theorem".
- Runge's_theorem subject Category:Theorems_in_complex_analysis.
- Runge's_theorem type Abstraction100002137.
- Runge's_theorem type Communication100033020.
- Runge's_theorem type Message106598915.
- Runge's_theorem type Proposition106750804.
- Runge's_theorem type Statement106722453.
- Runge's_theorem type Theorem106752293.
- Runge's_theorem type TheoremsInComplexAnalysis.
- Runge's_theorem comment "In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following: Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K.".
- Runge's_theorem label "Runge's theorem".
- Runge's_theorem label "Runge-Theorie".
- Runge's_theorem label "Théorème de Runge".
- Runge's_theorem label "Теорема Рунге".
- Runge's_theorem sameAs Runge-Theorie.
- Runge's_theorem sameAs Théorème_de_Runge.
- Runge's_theorem sameAs 룽에의_정리.
- Runge's_theorem sameAs m.08ysj_.
- Runge's_theorem sameAs Q1505529.
- Runge's_theorem sameAs Q1505529.
- Runge's_theorem sameAs Runge's_theorem.
- Runge's_theorem wasDerivedFrom Runge's_theorem?oldid=561421124.
- Runge's_theorem depiction Runge_theorem.svg.
- Runge's_theorem isPrimaryTopicOf Runge's_theorem.
