Liouville's theorem
E854242
Liouville's theorem is a fundamental result in complex analysis stating that any bounded entire function must be constant.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Liouville's theorem canonical | 3 |
| Liouville theorem | 1 |
| Liouville's theorem in complex analysis | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10304982 — 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: Liouville's theorem Context triple: [Cauchy integral formula, implies, Liouville's theorem]
-
A.
Liouville's theorem in Hamiltonian mechanics
Liouville's theorem in Hamiltonian mechanics states that the phase-space volume occupied by an ensemble of systems evolving under Hamiltonian dynamics is conserved over time, implying incompressible flow in phase space.
-
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.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
D.
Morera's theorem
Morera's theorem is a fundamental result in complex analysis that characterizes holomorphic functions by stating that a continuous function with zero integral over every closed contour in a domain must be analytic there.
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E.
Lindelöf theorem in complex analysis
The Lindelöf theorem in complex analysis is a result that refines the maximum modulus principle by controlling the boundary growth of analytic functions, particularly along paths approaching boundary points of their domain.
- 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: Liouville's theorem Target entity description: Liouville's theorem is a fundamental result in complex analysis stating that any bounded entire function must be constant.
-
A.
Liouville's theorem in Hamiltonian mechanics
Liouville's theorem in Hamiltonian mechanics states that the phase-space volume occupied by an ensemble of systems evolving under Hamiltonian dynamics is conserved over time, implying incompressible flow in phase space.
-
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.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
D.
Morera's theorem
Morera's theorem is a fundamental result in complex analysis that characterizes holomorphic functions by stating that a continuous function with zero integral over every closed contour in a domain must be analytic there.
-
E.
Lindelöf theorem in complex analysis
The Lindelöf theorem in complex analysis is a result that refines the maximum modulus principle by controlling the boundary growth of analytic functions, particularly along paths approaching boundary points of their domain.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Liouville's theorem Description of subject: Liouville's theorem is a fundamental result in complex analysis stating that any bounded entire function must be constant.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Liouville's theorem in complex analysis
this entity surface form:
Liouville theorem