Runge approximation theorem

E480873

approximation theorem theorem in complex analysis

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.

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All labels observed (3)

Label	Occurrences
Runge approximation theorem canonical	1
Runge theorem	1
Runge's theorem	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.
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
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 ⓘ Oka–Weil theorem ⓘ
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 ⓘ
provenBy	Carl Runge ⓘ
relatedTo	Mergelyan's theorem ⓘ Oka–Weil theorem ⓘ Stone–Weierstrass theorem ⓘ Weierstrass approximation theorem ⓘ
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 ⓘ linked to: Oka coherence theorem approximation theory ⓘ complex dynamical systems ⓘ construction of holomorphic functions with prescribed properties ⓘ function theory ⓘ theory of Riemann surfaces ⓘ linked to: Riemann surfaces

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Weierstrass approximation theorem → relatedTo → Runge approximation theorem ⓘ

Mittag-Leffler theorem → relatedTo → Runge's theorem ⓘ

linked to: Runge approximation theorem

Functions of One Complex Variable → topic → Runge theorem ⓘ

subject linked to: Functions of One Complex Variable by John B. Conway