Euler–Maclaurin summation formula
E54271
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Euler–Maclaurin summation formula canonical | 2 |
| Euler summation | 1 |
| Maclaurin | 1 |
| Stirling series | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T426770 — 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: Euler–Maclaurin summation formula Context triple: [Leonhard Euler, notableWork, Euler–Maclaurin summation formula]
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
D.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- 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: Euler–Maclaurin summation formula Target entity description: 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.
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
D.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in mathematical analysis ⓘ |
| appearsIn |
monographs on asymptotic expansions
ⓘ
textbooks on analytic number theory ⓘ treatises on numerical integration ⓘ |
| appliesTo | sufficiently smooth functions ⓘ |
| category |
asymptotic formula
ⓘ
summation formula ⓘ |
| field |
analytic number theory
ⓘ
asymptotic analysis ⓘ mathematical analysis ⓘ numerical analysis ⓘ |
| firstContributor | Colin Maclaurin ⓘ |
| furtherDevelopedBy | Leonhard Euler ⓘ |
| generalizes | trapezoidal rule error expansion ⓘ |
| hasComponent |
endpoint correction terms
ⓘ
integral term ⓘ remainder term ⓘ series of derivative terms ⓘ |
| historicalPeriod | 18th century ⓘ |
| namedAfter |
Colin Maclaurin
ⓘ
Leonhard Euler ⓘ |
| property |
gives exact equality when full infinite expansion is used under suitable conditions
ⓘ
remainder can often be bounded using higher derivatives ⓘ truncated expansion yields asymptotic approximation ⓘ |
| provides |
asymptotic expansions for sums
ⓘ
connection between sums and integrals ⓘ error estimates for approximating sums by integrals ⓘ |
| relatedTo |
Bernoulli numbers
ⓘ
surface form:
Faulhaber's formula
Poisson summation formula ⓘ Stirling's approximation ⓘ |
| relates |
definite integrals
ⓘ
finite sums ⓘ infinite series ⓘ |
| typicalAssumption |
derivatives of the function satisfy suitable decay or boundedness conditions
ⓘ
function has sufficiently many continuous derivatives on the interval ⓘ |
| usedFor |
accelerating convergence of series
ⓘ
approximating series by integrals ⓘ deriving Stirling-type approximations ⓘ deriving asymptotic expansions of sums ⓘ estimating truncation errors in numerical quadrature ⓘ evaluating slowly convergent series ⓘ |
| usedIn |
computation of the Riemann zeta function
ⓘ
lattice point counting problems ⓘ spectral methods in numerical analysis ⓘ |
| uses |
Bernoulli numbers
ⓘ
higher derivatives of a function ⓘ |
How these facts were elicited
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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: Euler–Maclaurin summation formula Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Euler summation
this entity surface form:
Maclaurin
this entity surface form:
Stirling series