Borel summation
E451516
Borel summation is a mathematical technique that assigns finite values to certain divergent series by transforming and analytically continuing their associated power series.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Borel summation canonical | 3 |
| Borel transform | 1 |
| Borel–Leroy summation | 1 |
| Borel–Écalle summation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4552233 — 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: Borel summation Context triple: [Divergent Series, topic, Borel summation]
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
C.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
D.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
E.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
- 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: Borel summation Target entity description: Borel summation is a mathematical technique that assigns finite values to certain divergent series by transforming and analytically continuing their associated power series.
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
C.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
D.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
E.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical technique
ⓘ
method of summability ⓘ summation method ⓘ |
| appliesTo |
asymptotic series
ⓘ
divergent power series ⓘ formal power series ⓘ |
| basedOn |
Borel transform
ⓘ
Laplace transform ⓘ |
| characterizedBy |
analytic continuation of Borel transform
ⓘ
integral representation ⓘ use of exponential damping factor ⓘ |
| compatibleWith | ordinary convergence of series ⓘ |
| field |
asymptotic analysis
ⓘ
complex analysis ⓘ functional analysis ⓘ mathematical analysis ⓘ summability theory ⓘ |
| generalizes | ordinary summation of convergent series ⓘ |
| hasVariant |
Borel–Leroy summation
ⓘ
linked to:
Borel summation
Borel–Écalle summation ⓘ
linked to:
Borel summation
iterated Borel summation ⓘ |
| historicalDevelopment |
developed by Émile Borel in the context of divergent series
ⓘ
introduced in early 20th century ⓘ |
| namedAfter | Émile Borel ⓘ |
| property |
can assign finite values to some factorially divergent series
ⓘ
linear summation method ⓘ regular summation method on convergent series ⓘ |
| relatedTo |
Laplace–Borel transform
ⓘ
linked to:
Laplace transform
Stokes phenomena ⓘ
linked to:
Stokes phenomenon
Watson’s lemma ⓘ
linked to:
Laplace method
resummation methods ⓘ Écalle resurgence ⓘ |
| requires |
analytic continuation along integration path
ⓘ
existence of Borel transform ⓘ suitable growth conditions at infinity ⓘ |
| strongerThan |
Abel summation
ⓘ
Cesàro summation ⓘ |
| usedFor |
analytic continuation of power series
ⓘ
assigning values to divergent series ⓘ regularization of divergent series ⓘ resummation of asymptotic expansions ⓘ |
| usedIn |
analytic number theory
ⓘ
differential equations ⓘ perturbation theory ⓘ quantum field theory ⓘ resurgence theory ⓘ theory of divergent integrals ⓘ |
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: Borel summation Description of subject: Borel summation is a mathematical technique that assigns finite values to certain divergent series by transforming and analytically continuing their associated power series.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Borel summation
linked to: Borel summation
linked to: Borel summation