Picard theorem

E326981

mathematical theorem theorem in complex analysis

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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All labels observed (9)

Label	Occurrences
Great Picard theorem	2
Little Picard theorem	2
Casorati–Weierstrass theorem (converse directions and related results)	1
Liouville's theorem	1
Picard theorem canonical	1
Picard's second theorem	1
Picard's theorem	1
Picard’s great theorem	1
value distribution theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T3115957 — 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: Picard theorem
Context triple: [Émile Picard, notableWork, Picard 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. 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.
C. Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
D. Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
E. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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: Picard theorem
Target entity description: 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.

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. 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.
C. Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
D. Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
E. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in complex analysis ⓘ
appliesTo	entire non-constant functions ⓘ functions holomorphic on ℂ ⓘ functions meromorphic near an essential singularity ⓘ
clarifies	behavior of entire functions at infinity ⓘ structure of essential singularities ⓘ
concerns	entire functions ⓘ holomorphic functions ⓘ image of holomorphic maps ⓘ meromorphic functions ⓘ
field	complex analysis ⓘ
hasAlternativeName	Great Picard theorem ⓘ linked to: Picard theorem Picard’s great theorem ⓘ linked to: Picard theorem
hasConsequence	essential singularities are dense in their neighborhoods in terms of image ⓘ near an essential singularity a function attains every complex value, with at most one exception, infinitely often ⓘ
hasExample	e^z is a non-constant entire function omitting exactly one value (0) ⓘ the exponential function omits the value 0 in ℂ* ⓘ
hasGeneralization	Ahlfors’ theory of covering surfaces ⓘ Nevanlinna’s value distribution theory ⓘ linked to: Nevanlinna theory
hasHistoricalPeriod	late 19th century ⓘ
hasImportance	cornerstone of value distribution theory ⓘ fundamental result in complex analysis ⓘ
hasKeyConcept	entire non-polynomial functions ⓘ essential singularity ⓘ omitted values ⓘ
hasProofMethod	Montel theorem ⓘ Riemann surface theory ⓘ linked to: Riemann surfaces normal families ⓘ potential theory ⓘ
hasScope	functions defined on the complex plane ⓘ
hasVariant	Great Picard theorem ⓘ linked to: Picard theorem Little Picard theorem ⓘ linked to: Picard theorem
implies	a holomorphic map from ℂ to the Riemann sphere minus two points is constant ⓘ a non-constant entire function cannot omit a non-empty open subset of ℂ ⓘ an entire function omitting two distinct complex values is constant ⓘ if an entire function omits an open set, it is constant ⓘ the image of a non-constant entire function is either all of ℂ or ℂ minus one point ⓘ
isStrongerThan	Liouville theorem ⓘ linked to: Liouville's theorem
namedAfter	Émile Picard ⓘ
relatedTo	Casorati–Weierstrass theorem ⓘ Riemann sphere ⓘ entire transcendental functions ⓘ
states	a non-constant entire function takes every complex value, with at most one exception ⓘ
usedIn	Nevanlinna theory ⓘ value distribution theory ⓘ
usedInProofOf	Casorati–Weierstrass theorem (converse directions and related results) ⓘ linked to: Picard theorem

How these facts were elicited

Referenced by (11)

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

Émile Picard → notableWork → Picard's second theorem ⓘ

linked to: Picard theorem

Émile Picard → notableWork → Picard theorem ⓘ

Cauchy integral theorem → implies → Liouville's theorem ⓘ

linked to: Picard theorem

Schwarz lemma → relatedResult → Little Picard theorem ⓘ

linked to: Picard theorem

Montel's theorem → relatedTo → Picard's theorem ⓘ

subject linked to: Montel theorem

linked to: Picard theorem

Georges Valiron → contributedTo → value distribution theory ⓘ

linked to: Picard theorem

Picard theorem → hasAlternativeName → Great Picard theorem ⓘ

linked to: Picard theorem

Picard theorem → hasAlternativeName → Picard’s great theorem ⓘ

linked to: Picard theorem

Picard theorem → hasVariant → Little Picard theorem ⓘ

linked to: Picard theorem

Picard theorem → hasVariant → Great Picard theorem ⓘ

linked to: Picard theorem

Picard theorem → usedInProofOf → Casorati–Weierstrass theorem (converse directions and related results) ⓘ

linked to: Picard theorem

All labels observed (9)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (11)

How this entity was disambiguated

How these facts were elicited