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.

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All labels observed (1)

Label	Occurrences
Rouché's theorem canonical	1

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 ⓘ

Referenced by (1)

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

Cauchy residue theorem → isRelatedTo → Rouché's theorem ⓘ