Euler-Maclaurin formula

For more general approximations, see the Euler-Maclaurin formula.

The Euler-Maclaurin formula is also used for detailed error analysis in numerical quadrature.

Euler computed this sum to 20 decimal places with only a few terms of the Euler-Maclaurin formula in 1735.

The most notable of these approximations are Euler's method and the Euler-Maclaurin formula.

In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums.

Independently from Euler and using the same methods, Maclaurin discovered the Euler-Maclaurin formula.

The Euler-MacLaurin formula can be understood as a curious application of some ideas from Banach spaces and functional analysis.

Arguably the most important application of the Bernoulli number in mathematics is their use in the Euler-MacLaurin formula.

Similarly, the Euler-MacLaurin formula cannot be applied either, as it, too, expresses an theorem ultimately anchored in the theory of finite differences.

June 8 - Leonhard Euler writes to James Stirling describing the Euler-Maclaurin formula, providing a connection between integrals and calculus.

Methods of generating such expansions include the Euler-Maclaurin formula and integral transforms such as the Laplace and Mellin transforms.

Essentially, Euler-MacLaurin summation can be applied whenever Carlson's theorem holds; the Euler-MacLaurin formula is essentially a result obtaining from the study of finite differences and Newton series.

A periodic Bernoulli polynomial P(x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler-Maclaurin formula relating sums to integrals.

Clenshaw-Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler-Maclaurin approach is very accurate (in that particular case the Euler-Maclaurin formula takes the form of a discrete cosine transform).

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler-Maclaurin formula, and in expressions for certain values of the Riemann zeta function.