Weierstrass factorization theorem
E110607
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hadamard factorization theorem | 3 |
| Weierstrass factorization theorem canonical | 1 |
| Weierstrass primary factors | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T940260 — 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: Weierstrass factorization theorem Context triple: [Karl Weierstrass, notableFor, Weierstrass factorization theorem]
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A.
Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
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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.
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C.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
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D.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
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E.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
- 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: Weierstrass factorization theorem Target entity description: 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.
-
A.
Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
-
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.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
D.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
E.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in complex analysis ⓘ |
| allows | prescribing zeros of an entire function with given multiplicities ⓘ |
| appliesTo | entire functions ⓘ |
| assumes | zeros form a discrete subset of the complex plane ⓘ |
| codomain | representations as infinite products ⓘ |
| conclusion |
entire function equals e^{g(z)} times a canonical product over its zeros for some entire g
ⓘ
entire function is determined up to a nonvanishing entire factor without zeros ⓘ |
| context | complex plane ⓘ |
| describes | factorization of entire functions ⓘ |
| domain | functions from complex numbers to complex numbers ⓘ |
| ensures | convergence of infinite products via suitable exponential factors ⓘ |
| field |
complex analysis
ⓘ
mathematical analysis ⓘ |
| generalizes | factorization of polynomials into linear factors ⓘ |
| hasApplicationIn |
analytic number theory
ⓘ
construction of special entire functions ⓘ functional analysis ⓘ theory of meromorphic functions ⓘ |
| hasFormulation |
for entire functions with zeros of finite multiplicity
ⓘ
in terms of canonical products of minimal genus ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| implies | existence of entire functions with arbitrary prescribed discrete zero sets without accumulation in the finite plane ⓘ |
| influenced | development of modern function theory ⓘ |
| involves |
infinite products
ⓘ
zeros of entire functions ⓘ |
| isIncludedIn | standard complex analysis textbooks ⓘ |
| isPartOf | classical theory of entire functions ⓘ |
| isRelatedTo |
Weierstrass factorization theorem
self-linksurface differs
ⓘ
surface form:
Hadamard factorization theorem
Mittag-Leffler theorem ⓘ canonical product of genus p ⓘ infinite product representations of analytic functions ⓘ order of an entire function ⓘ |
| isTaughtIn |
advanced undergraduate complex analysis courses
ⓘ
graduate complex analysis courses ⓘ |
| language | mathematical notation ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| requires |
basic complex function theory
ⓘ
canonical products ⓘ knowledge of convergence of infinite products ⓘ |
| statesThat | every entire function can be represented as an infinite product determined by its zeros ⓘ |
| usedFor |
building entire functions with prescribed growth and zeros
ⓘ
product representations of the Riemann zeta function ⓘ product representations of the gamma function ⓘ proving properties of special functions such as the sine function ⓘ |
| uses |
Weierstrass factorization theorem
self-linksurface differs
ⓘ
surface form:
Weierstrass primary factors
|
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Weierstrass factorization theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hadamard factorization theorem
Weierstrass factorization theorem
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uses
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Weierstrass factorization theorem
self-linksurface differs
ⓘ
this entity surface form:
Weierstrass primary factors
Weierstrass factorization theorem
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isRelatedTo
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Weierstrass factorization theorem
self-linksurface differs
ⓘ
this entity surface form:
Hadamard factorization theorem
Essai sur l’étude des fonctions données par leur développement de Taylor
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relatedConcept
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Weierstrass factorization theorem
ⓘ
this entity surface form:
Hadamard factorization theorem