Morera's theorem
E825426
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Morera's theorem canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T9843561 — 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: Morera's theorem Context triple: [Cauchy integral theorem, implies, Morera's theorem]
-
A.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
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B.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
C.
Cauchy integral theorem
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.
-
D.
Lindelöf theorem in complex analysis
The Lindelöf theorem in complex analysis is a result that refines the maximum modulus principle by controlling the boundary growth of analytic functions, particularly along paths approaching boundary points of their domain.
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E.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
- 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: Morera's theorem Target entity description: 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.
-
A.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
B.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
C.
Cauchy integral theorem
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.
-
D.
Lindelöf theorem in complex analysis
The Lindelöf theorem in complex analysis is a result that refines the maximum modulus principle by controlling the boundary growth of analytic functions, particularly along paths approaching boundary points of their domain.
-
E.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
- F. None of above. chosen
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 ⓘ |
| 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 ⓘ |
| proofTechnique | approximation of curves by polygonal paths or triangles ⓘ |
| relatedTo |
Cauchy integral formula
ⓘ
Cauchy integral theorem ⓘ Goursat's theorem ⓘ
linked to:
Cauchy integral theorem
|
| 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 ⓘ |
| usesConcept |
Cauchy integral theorem
ⓘ
closed contour ⓘ contour integral ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Morera's theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.