Runge approximation theorem
E480873
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Runge approximation theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4927205 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Runge approximation theorem Context triple: [Weierstrass approximation theorem, relatedTo, Runge approximation theorem]
-
A.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
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B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
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C.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
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D.
Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
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E.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Runge approximation theorem Target entity description: The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
A.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
C.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
D.
Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
-
E.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
approximation theorem
ⓘ
theorem in complex analysis ⓘ |
| appearsIn |
courses on several complex variables
ⓘ
textbooks on complex analysis ⓘ |
| appliesTo |
holomorphic functions on open sets
ⓘ
open subsets of the complex plane ⓘ |
| assumes |
holomorphicity on an open neighborhood of the compact set
ⓘ
topological conditions on the complement of the compact set ⓘ |
| codomain | complex numbers ⓘ |
| concerns |
approximation in the supremum norm
ⓘ
uniform convergence on compact sets ⓘ |
| dealsWith |
domains in the complex plane
ⓘ
holomorphic functions ⓘ poles of rational functions ⓘ rational functions ⓘ uniform approximation ⓘ |
| field |
complex analysis
ⓘ
mathematical analysis ⓘ |
| generalizedBy |
Mergelyan's theorem
NERFINISHED
ⓘ
Oka–Weil theorem NERFINISHED ⓘ |
| hasVersion |
polynomial approximation version
ⓘ
rational approximation version ⓘ |
| historicalPeriod | late 19th century ⓘ |
| implies |
polynomials are dense in the space of holomorphic functions on compact sets with connected complement
ⓘ
rational functions with prescribed poles are dense in spaces of holomorphic functions on suitable compact sets ⓘ |
| involvesConcept |
compact subsets of the complex plane
ⓘ
components of the complement ⓘ connected complement ⓘ density of subalgebras of holomorphic functions ⓘ holomorphic extension to neighborhoods ⓘ |
| isToolFor |
approximating holomorphic maps by simpler functions
ⓘ
constructing holomorphic functions avoiding given sets of poles ⓘ |
| mathematicalSubjectClassification | 30E10 ⓘ |
| namedAfter | Carl Runge NERFINISHED ⓘ |
| provenBy | Carl Runge NERFINISHED ⓘ |
| relatedTo |
Mergelyan's theorem
NERFINISHED
ⓘ
Oka–Weil theorem NERFINISHED ⓘ Stone–Weierstrass theorem NERFINISHED ⓘ Weierstrass approximation theorem NERFINISHED ⓘ |
| statesThat |
holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains
ⓘ
if K is a compact subset of C and A is a set containing at most one point from each component of C \ K then every function holomorphic on a neighborhood of K can be uniformly approximated on K by rational functions with poles in A ⓘ if K is a compact subset of C with connected complement and f is holomorphic on an open set containing K then f can be uniformly approximated on K by polynomials ⓘ |
| typicalDomain | subset of the complex plane C ⓘ |
| usedIn |
Oka theory
NERFINISHED
ⓘ
approximation theory ⓘ complex dynamical systems ⓘ construction of holomorphic functions with prescribed properties ⓘ function theory ⓘ theory of Riemann surfaces NERFINISHED ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Runge approximation theorem Description of subject: The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.