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's theorem in complex analysis | 2 |
| Liouville theorem | 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.
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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.
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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.
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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 ⓘ |
| proofMethod |
Cauchy estimates
ⓘ
linked to:
Cauchy integral formula
Cauchy integral formula ⓘ maximum modulus principle ⓘ |
| quantification | If f:ℂ→ℂ is entire and bounded, then f is constant ⓘ |
| relatedTo |
Cauchy integral formula
ⓘ
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Liouville's theorem
linked to: Liouville's theorem
Liouville's theorem in Hamiltonian mechanics
→
relatedTo
→
Liouville's theorem in complex analysis
ⓘ
linked to: Liouville's theorem