Liouville's theorem

E854242

theorem in complex analysis

Liouville's theorem is a fundamental result in complex analysis stating that any bounded entire function must be constant.

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Observed surface forms (3)

Surface form	Occurrences
Liouville theorem	1
Liouville's theorem in complex analysis	1
Liouville's theorem (complex analysis)	0

Statements (41)

Predicate	Object
instanceOf	theorem in complex analysis ⓘ
appliesTo	entire functions ⓘ holomorphic functions on ℂ ⓘ
assumption	complex differentiability at every point of ℂ ⓘ existence of a global bound on the modulus of the function ⓘ
boundednessCondition	function is bounded on the complex plane ⓘ
category	result about global behavior of holomorphic functions ⓘ
conclusion	function must be constant ⓘ
consequence	entire functions with bounded real part are constant (via related results) ⓘ
contrastWith	existence of non-constant bounded holomorphic functions on proper domains of ℂ (e.g. unit disk) ⓘ
corollaryOf	Cauchy integral formula and Cauchy estimates ⓘ
domainCondition	function is defined on the whole complex plane ⓘ function is entire ⓘ
field	complex analysis ⓘ
generalizationOf	the fact that bounded harmonic entire functions are constant ⓘ
hasGeneralization	Liouville-type theorems for harmonic functions ⓘ Liouville-type theorems in several complex variables ⓘ
historicalPeriod	19th century mathematics ⓘ
holdsIn	complex plane ⓘ
implies	no non-constant entire function can be uniformly bounded on ℂ ⓘ non-constant entire functions are unbounded ⓘ polynomials that are bounded on the complex plane are constant polynomials ⓘ
isEquivalentTo	Every bounded entire function attains its maximum modulus everywhere only if it is constant ⓘ
level	undergraduate complex analysis ⓘ
namedAfter	Joseph Liouville NERFINISHED ⓘ
proofMethod	Cauchy estimates NERFINISHED ⓘ Cauchy integral formula NERFINISHED ⓘ maximum modulus principle ⓘ
quantification	If f:ℂ→ℂ is entire and bounded, then f is constant ⓘ
relatedTo	Cauchy integral formula NERFINISHED ⓘ fundamental theorem of algebra ⓘ maximum modulus principle ⓘ
requires	basic properties of complex differentiability ⓘ notions of holomorphic and entire functions ⓘ
statement	Every bounded entire function is constant ⓘ
typeOfResult	rigidity theorem ⓘ
usedFor	classifying bounded holomorphic functions on the complex plane ⓘ proving the fundamental theorem of algebra ⓘ showing that entire functions with certain growth restrictions are polynomials ⓘ
usedIn	complex dynamics ⓘ function theory on the complex plane ⓘ

Referenced by (5)

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

Cauchy integral formula → implies → Liouville's theorem ⓘ

Joseph Liouville → notableWork → Liouville's theorem ⓘ

this entity surface form: Liouville's theorem in complex analysis

Joseph Liouville → hasEponym → Liouville's theorem ⓘ

Picard theorem → isStrongerThan → Liouville's theorem ⓘ

this entity surface form: Liouville theorem

Hadamard three-circle theorem → contrastWith → Liouville's theorem ⓘ