Cauchy integral theorem

E239284

result in mathematical analysis theorem in complex analysis

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.

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All labels observed (6)

Label	Occurrences
Cauchy integral theorem canonical	10
Cauchy integral theorem (by analogy)	1
Cauchy–Goursat theorem	1
Cauchy’s integral theorem	1
Goursat's 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.
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
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

Referenced by (15)

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

Augustin-Louis Cauchy → knownFor → Cauchy integral theorem ⓘ

Augustin-Louis Cauchy → notableFor → Cauchy integral theorem ⓘ

subject linked to: Augustin-Louis

Cauchy integral theorem → hasVersion → classical Cauchy integral theorem ⓘ