Weierstrass factorization theorem

E110607

mathematical theorem theorem in complex analysis

The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.

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All labels observed (3)

Label	Occurrences
Hadamard factorization theorem	3
Weierstrass factorization theorem canonical	1
Weierstrass primary factors	1

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in complex analysis ⓘ
allows	prescribing zeros of an entire function with given multiplicities ⓘ
appliesTo	entire functions ⓘ
assumes	zeros form a discrete subset of the complex plane ⓘ
codomain	representations as infinite products ⓘ
conclusion	entire function equals e^{g(z)} times a canonical product over its zeros for some entire g ⓘ entire function is determined up to a nonvanishing entire factor without zeros ⓘ
context	complex plane ⓘ
describes	factorization of entire functions ⓘ
domain	functions from complex numbers to complex numbers ⓘ
ensures	convergence of infinite products via suitable exponential factors ⓘ
field	complex analysis ⓘ mathematical analysis ⓘ
generalizes	factorization of polynomials into linear factors ⓘ
hasApplicationIn	analytic number theory ⓘ construction of special entire functions ⓘ functional analysis ⓘ theory of meromorphic functions ⓘ
hasFormulation	for entire functions with zeros of finite multiplicity ⓘ in terms of canonical products of minimal genus ⓘ
historicalPeriod	19th century mathematics ⓘ
implies	existence of entire functions with arbitrary prescribed discrete zero sets without accumulation in the finite plane ⓘ
influenced	development of modern function theory ⓘ
involves	infinite products ⓘ zeros of entire functions ⓘ
isIncludedIn	standard complex analysis textbooks ⓘ
isPartOf	classical theory of entire functions ⓘ
isRelatedTo	Weierstrass factorization theorem self-linksurface differs ⓘ surface form: Hadamard factorization theorem Mittag-Leffler theorem ⓘ canonical product of genus p ⓘ infinite product representations of analytic functions ⓘ order of an entire function ⓘ
isTaughtIn	advanced undergraduate complex analysis courses ⓘ graduate complex analysis courses ⓘ
language	mathematical notation ⓘ
namedAfter	Karl Weierstrass ⓘ
requires	basic complex function theory ⓘ canonical products ⓘ knowledge of convergence of infinite products ⓘ
statesThat	every entire function can be represented as an infinite product determined by its zeros ⓘ
usedFor	building entire functions with prescribed growth and zeros ⓘ product representations of the Riemann zeta function ⓘ product representations of the gamma function ⓘ proving properties of special functions such as the sine function ⓘ
uses	Weierstrass factorization theorem self-linksurface differs ⓘ surface form: Weierstrass primary factors

Referenced by (5)

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

Karl Weierstrass → notableFor → Weierstrass factorization theorem ⓘ

Jacques Hadamard → knownFor → Weierstrass factorization theorem ⓘ

this entity surface form: Hadamard factorization theorem

Weierstrass factorization theorem → uses → Weierstrass factorization theorem self-linksurface differs ⓘ

this entity surface form: Weierstrass primary factors

Weierstrass factorization theorem → isRelatedTo → Weierstrass factorization theorem self-linksurface differs ⓘ

this entity surface form: Hadamard factorization theorem

Essai sur l’étude des fonctions données par leur développement de Taylor → relatedConcept → Weierstrass factorization theorem ⓘ

this entity surface form: Hadamard factorization theorem