Cauchy–Hadamard theorem
E239292
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cauchy–Hadamard theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171653 — 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: Cauchy–Hadamard theorem Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy–Hadamard theorem]
-
A.
Weierstrass factorization theorem
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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B.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
C.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
D.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
E.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
- 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: Cauchy–Hadamard theorem Target entity description: The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
A.
Weierstrass factorization theorem
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.
-
B.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
C.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
D.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
E.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in complex analysis ⓘ |
| appearsIn |
advanced undergraduate mathematics curriculum
ⓘ
graduate complex analysis textbooks ⓘ |
| appliesTo |
complex power series
ⓘ
formal power series with complex coefficients ⓘ |
| assumes | power series ∑ a_n z^n ⓘ |
| category |
theorem about analytic functions
ⓘ
theorem about series ⓘ |
| characterizes | radius of convergence of a power series ⓘ |
| concerns | disks of convergence in the complex plane ⓘ |
| concludes |
if limsup_{n→∞} |a_n|^{1/n} = 0 then radius of convergence is ∞
ⓘ
if limsup_{n→∞} |a_n|^{1/n} = ∞ then radius of convergence is 0 ⓘ radius of convergence R = 1 / (limsup_{n→∞} |a_n|^{1/n}) ⓘ |
| context | local behavior of holomorphic functions ⓘ |
| doesNotDetermine | behavior on boundary |z| = R ⓘ |
| field | complex analysis ⓘ |
| generalizationOf | root test for series of real numbers ⓘ |
| givesFormulaFor | radius of convergence ⓘ |
| hasAlternativeFormulation | log R = - limsup_{n→∞} (1/n) log |a_n| when a_n ≠ 0 ⓘ |
| hasFormulation | R^{-1} = limsup_{n→∞} |a_n|^{1/n} ⓘ |
| holdsOver |
complete valued fields with absolute value
ⓘ
complex numbers ⓘ |
| implies |
power series converges absolutely inside disk |z| < R
ⓘ
power series diverges for |z| > R ⓘ |
| importance | fundamental result in complex analysis ⓘ |
| influencedBy |
Hadamard's work on series and entire functions
ⓘ
work of Cauchy on power series ⓘ |
| language | usually stated in terms of complex variable z ⓘ |
| namedAfter |
Augustin-Louis Cauchy
ⓘ
Jacques Hadamard ⓘ |
| proofUses |
estimates on |a_n z^n|
ⓘ
properties of limsup ⓘ root test for series ⓘ |
| relatedTo |
analytic functions
ⓘ
ratio test ⓘ root test ⓘ |
| relates | radius of convergence to growth of coefficients ⓘ |
| statementForm | R = 1 / L where L is limsup of |a_n|^{1/n} ⓘ |
| subject |
power series
ⓘ
radius of convergence ⓘ |
| usedAs |
criterion for convergence of power series
ⓘ
tool in estimating growth of analytic functions ⓘ |
| usedIn |
determining domains of convergence
ⓘ
study of Taylor series of holomorphic functions ⓘ theory of analytic continuation ⓘ |
| usesConcept |
complex variable
ⓘ
limsup ⓘ sequence of coefficients ⓘ |
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: Cauchy–Hadamard theorem Description of subject: The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.