Cauchy residue theorem

E239298

result in complex analysis theorem

The Cauchy residue theorem is a fundamental result in complex analysis that relates contour integrals of analytic functions around singularities to the sum of their residues, greatly simplifying the evaluation of many complex and real integrals.

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All labels observed (2)

Label	Occurrences
residue theorem	2
Cauchy residue theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T2171660 — 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 residue theorem
Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy residue 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-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
C. Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
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. 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.
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 residue theorem
Target entity description: The Cauchy residue theorem is a fundamental result in complex analysis that relates contour integrals of analytic functions around singularities to the sum of their residues, greatly simplifying the evaluation of many complex and real integrals.

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-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
C. Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
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. 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.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	result in complex analysis ⓘ theorem ⓘ
appliesIn	asymptotic analysis of integrals ⓘ electromagnetic theory ⓘ evaluation of Fourier integrals ⓘ evaluation of Laplace-type integrals ⓘ quantum field theory calculations ⓘ signal processing ⓘ
appliesTo	analytic functions with isolated singularities ⓘ meromorphic functions ⓘ
assumes	orientation of contour is positive (counterclockwise) in standard form ⓘ
concerns	isolated singularities ⓘ
field	complex analysis ⓘ
formula	∮_γ f(z) dz = 2πi Σ Res(f, a_k) for singularities a_k inside γ ⓘ
generalizes	Cauchy integral formula ⓘ
hasConsequence	integral around a closed contour is zero if there are no singularities inside ⓘ sum of residues including at infinity can be zero in certain settings ⓘ
hasExtension	global residue theorem in several complex variables ⓘ residue theorem on Riemann surfaces ⓘ
hasVariant	version with winding number n(γ, a_k) ⓘ
historicalPeriod	19th century mathematics ⓘ
implies	Cauchy integral theorem ⓘ
isFoundationFor	many contour integration techniques ⓘ residue calculus ⓘ
isRelatedTo	Jordan's lemma ⓘ Laurent series ⓘ Rouché's theorem ⓘ argument principle ⓘ
isTaughtIn	graduate analysis courses ⓘ undergraduate complex analysis courses ⓘ
isToolFor	evaluating complex integrals ⓘ evaluating improper integrals ⓘ evaluating integrals involving exponential functions ⓘ evaluating integrals involving rational functions ⓘ evaluating integrals involving trigonometric functions ⓘ evaluating real integrals via contour integration ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
relates	contour integrals ⓘ residues ⓘ
requires	contour to be closed ⓘ finite number of singularities inside the contour ⓘ function to be analytic on and inside the contour except at isolated singularities ⓘ
statesThat	the integral of a function around a closed contour equals 2πi times the sum of residues inside the contour ⓘ
usesConcept	Laurent series expansion ⓘ closed contour ⓘ residue of a complex function ⓘ winding number ⓘ
variantFormula	∮_γ f(z) dz = 2πi Σ n(γ, a_k) Res(f, a_k) ⓘ

How these facts were elicited

Referenced by (3)

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

Augustin-Louis Cauchy → knownFor → Cauchy residue theorem ⓘ

Cauchy integral theorem → relatedTo → Cauchy residue theorem ⓘ

this entity surface form: residue theorem

Cauchy integral formula → relatedTo → Cauchy residue theorem ⓘ