Morera's theorem

E825426

theorem in complex analysis

Morera's theorem is a fundamental result in complex analysis that characterizes holomorphic functions by stating that a continuous function with zero integral over every closed contour in a domain must be analytic there.

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Statements (44)

Predicate	Object
instanceOf	theorem in complex analysis ⓘ
appearsIn	graduate complex analysis courses ⓘ undergraduate complex analysis courses ⓘ
appliesTo	continuous complex-valued functions ⓘ
assumption	function is continuous on the domain ⓘ integral over every triangle in the domain is zero (common variant) ⓘ
characterizes	holomorphic functions ⓘ
conclusion	the function is analytic on the domain ⓘ the function is holomorphic on the domain ⓘ
condition	the integral of the function over every closed contour in the domain is zero ⓘ
doesNotRequire	prior existence of complex derivative ⓘ
domainOfApplication	open subsets of the complex plane ⓘ
equivalentFormulation	if the integral of a continuous function over every closed triangle in a domain is zero, then the function is holomorphic ⓘ
field	complex analysis ⓘ
generalizationOf	characterizations of holomorphic functions via vanishing integrals on special families of curves ⓘ
givesCriterionFor	analyticity ⓘ holomorphy ⓘ
hasGeneralization	Morera-type theorems in several complex variables ⓘ Morera-type theorems on Riemann surfaces NERFINISHED ⓘ
historicalPeriod	late 19th century ⓘ
holdsIn	general domains in the complex plane ⓘ simply connected domains ⓘ
implies	existence of complex derivative at every point of the domain ⓘ function satisfies Cauchy–Riemann equations (in the interior) ⓘ
language	mathematical logic and analysis ⓘ
logicalRole	converse to Cauchy integral theorem (up to continuity assumption) ⓘ
mathematicalSubjectClassification	30-XX (functions of a complex variable) ⓘ
namedAfter	Giuseppe Morera NERFINISHED ⓘ
proofTechnique	approximation of curves by polygonal paths or triangles ⓘ
relatedTo	Cauchy integral formula NERFINISHED ⓘ Cauchy integral theorem NERFINISHED ⓘ Goursat's theorem NERFINISHED ⓘ
requires	Green's theorem in typical proofs ⓘ continuity of the function on the domain ⓘ
standardReference	textbooks on complex analysis ⓘ
statementStyle	if-and-only-if characterization of holomorphicity via contour integrals ⓘ
typeOfCriterion	integral criterion ⓘ
usedFor	proving that a function is holomorphic without computing derivatives ⓘ showing locally uniform limits of holomorphic functions are holomorphic ⓘ showing uniform limits of holomorphic functions are holomorphic ⓘ
usedInProofOf	Weierstrass theorem on uniform limits of holomorphic functions NERFINISHED ⓘ
usesConcept	Cauchy integral theorem NERFINISHED ⓘ closed contour ⓘ contour integral ⓘ

Referenced by (3)

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

Cauchy integral theorem → implies → Morera's theorem ⓘ

Cauchy integral theorem → relatedTo → Morera's theorem ⓘ

Cauchy integral formula → relatedTo → Morera's theorem ⓘ