Rouché's theorem

E825437

theorem in complex analysis

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (2)

Label	Occurrences
Rouché theorem	1
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 ⓘ complex plane ⓘ
implies	small perturbations of a function do not change the number of zeros inside a contour ⓘ
language	mathematical English ⓘ
namedAfter	Eugène Rouché ⓘ
namedEntityType	mathematical theorem ⓘ
originalLanguage	French ⓘ
proofUses	argument principle ⓘ winding number ⓘ
relatedTo	Cauchy integral formula ⓘ Cauchy integral theorem ⓘ Fundamental Theorem of Algebra ⓘ Hurwitz's theorem ⓘ linked to: Hurwitz theorem (composition algebras) 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 ⓘ
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 ⓘ