Rouché's theorem
E825437
Rouché's theorem is a result in complex analysis that provides conditions under which two holomorphic functions have the same number of zeros inside a given contour.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Rouché's theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9844225 — 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: Rouché's theorem Context triple: [Cauchy residue theorem, isRelatedTo, Rouché's theorem]
-
A.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
B.
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.
-
C.
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.
-
D.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
E.
Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
- 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: Rouché's theorem Target entity description: Rouché's theorem is a result in complex analysis that provides conditions under which two holomorphic functions have the same number of zeros inside a given contour.
-
A.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
B.
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.
-
C.
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.
-
D.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
E.
Schwarz lemma
Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf | theorem in complex analysis ⓘ |
| appearsIn |
graduate-level complex analysis courses
ⓘ
textbooks on complex analysis ⓘ |
| appliesTo |
holomorphic functions on open subsets of the complex plane
ⓘ
meromorphic functions ⓘ |
| assumption |
contour is positively oriented
ⓘ
functions have no poles inside the contour (for holomorphic version) ⓘ |
| category | theorems about zeros of analytic functions ⓘ |
| conclusion |
f and g have the same number of zeros inside the contour
ⓘ
zeros counted with multiplicity ⓘ |
| coreCondition | |f(z) - g(z)| < |f(z)| on the contour ⓘ |
| domain | holomorphic functions ⓘ |
| field | complex analysis ⓘ |
| generalizationOf | results on stability of polynomial roots ⓘ |
| holdsIn |
Riemann surfaces
NERFINISHED
ⓘ
complex plane ⓘ |
| implies | small perturbations of a function do not change the number of zeros inside a contour ⓘ |
| language | mathematical English ⓘ |
| namedAfter | Eugène Rouché NERFINISHED ⓘ |
| namedEntityType | mathematical theorem ⓘ |
| originalLanguage | French ⓘ |
| proofUses |
argument principle
ⓘ
winding number ⓘ |
| relatedTo |
Cauchy integral formula
NERFINISHED
ⓘ
Cauchy integral theorem NERFINISHED ⓘ Fundamental Theorem of Algebra NERFINISHED ⓘ Hurwitz's theorem NERFINISHED ⓘ argument principle ⓘ maximum modulus principle ⓘ |
| requires |
closed contour
ⓘ
holomorphic functions on and inside the contour ⓘ simple closed contour ⓘ |
| statementForm | inequality on the boundary of a domain ⓘ |
| type |
localization theorem
ⓘ
zero-counting theorem NERFINISHED ⓘ |
| typicalApplication |
comparing a polynomial with its dominant term on a large circle
ⓘ
showing all roots of a polynomial lie in a given disk ⓘ |
| usedFor |
counting zeros of holomorphic functions
ⓘ
locating zeros of polynomials ⓘ proving the Fundamental Theorem of Algebra ⓘ root localization in numerical analysis ⓘ stability of zeros under perturbations ⓘ |
| yearIntroduced | 1862 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Rouché's theorem Description of subject: Rouché's theorem is a result in complex analysis that provides conditions under which two holomorphic functions have the same number of zeros inside a given contour.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.