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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.