Cauchy integral theorem
E239284
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cauchy integral theorem canonical | 4 |
| Cauchy–Goursat theorem | 1 |
| classical Cauchy integral theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171644 — 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: Cauchy integral theorem Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy integral theorem]
-
A.
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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B.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
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C.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
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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.
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E.
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.
- 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: Cauchy integral theorem Target entity description: The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
-
A.
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.
-
B.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
C.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
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.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in mathematical analysis
ⓘ
theorem in complex analysis ⓘ |
| appliesTo |
holomorphic functions on open subsets of the complex plane
ⓘ
holomorphic functions on simply connected domains ⓘ |
| assumes |
contour is closed
ⓘ
domain is simply connected in its basic form ⓘ function is holomorphic on an open set containing the contour and its interior ⓘ |
| characterizes | holomorphic functions via vanishing integrals over closed curves ⓘ |
| conclusion | integral of the function over the closed contour is zero ⓘ |
| coreStatement | the integral of a holomorphic function over any closed contour in a simply connected domain is zero ⓘ |
| dealsWith |
closed curves in the complex plane
ⓘ
contour integrals ⓘ holomorphic functions ⓘ |
| field |
complex analysis
ⓘ
mathematics ⓘ |
| generalizationOf | fundamental theorem of calculus to complex functions ⓘ |
| hasGeneralization |
versions for several complex variables
ⓘ
versions in differential forms language ⓘ |
| hasVersion |
Cauchy–Goursat theorem
ⓘ
linked to:
Cauchy integral theorem
classical Cauchy integral theorem ⓘ
linked to:
Cauchy integral theorem
homology version of Cauchy integral theorem ⓘ version for piecewise smooth closed curves ⓘ |
| historicalPeriod | 19th century ⓘ |
| holdsIn |
complex plane
ⓘ
open subsets of the complex plane ⓘ |
| implies |
Cauchy integral formula
ⓘ
Liouville's theorem ⓘ
linked to:
Picard theorem
Morera's theorem ⓘ existence of antiderivatives for holomorphic functions on simply connected domains ⓘ fundamental theorem of algebra ⓘ path independence of integrals of holomorphic functions in simply connected domains ⓘ |
| mathematicalDomain | theory of analytic functions ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| relatedTo |
Cauchy estimates
ⓘ
linked to:
Cauchy integral formula
Cauchy integral formula ⓘ Green's theorem ⓘ Morera's theorem ⓘ residue theorem ⓘ
linked to:
Cauchy residue theorem
|
| requires | complex differentiability on an open set containing the curve and its interior ⓘ |
| standardReference |
Complex Analysis by Elias M. Stein and Rami Shakarchi
ⓘ
linked to:
Complex Analysis
Complex Analysis by Lars Ahlfors ⓘ
linked to:
Complex Analysis
Functions of One Complex Variable by John B. Conway ⓘ |
| usedFor |
deriving estimates for derivatives of holomorphic functions
ⓘ
establishing power series expansions ⓘ evaluating complex integrals ⓘ proving properties of analytic functions ⓘ |
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: Cauchy integral theorem Description of subject: The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Augustin-Louis
linked to: Cauchy integral theorem
linked to: Cauchy integral theorem