Euler-Maclaurin formula


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Euler-Maclaurin formula

[¦ȯi·lər mə′klȯr·ən ‚fȯr·myə·lə]
(mathematics)
A formula used in the numerical evaluation of integrals, which states that the value of an integral is equal to the sum of the value given by the trapezoidal rule and a series of terms involving the odd-numbered derivatives of the function at the end points of the interval over which the integral is evaluated.
References in periodicals archive ?
For a general d, Zhao [12] proved the analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of d variables using the theory of generalized function and Akiyama, Egami and Tanigawa [1] proved the same result by applying the classical Euler-Maclaurin formula to the index of the summation [n.sub.d].
In this note we have illustrated applications of the Euler-Maclaurin formula to the estimation of graph entropies of paths and rings.
[9] __, The Euler-Maclaurin formula and sums of powers revisited, Int.
Balanzario, "A generalized Euler-Maclaurin formula for the Hurwitz zeta function," Mathematica Slovaca, vol.
In order to evaluate this function, we use the Euler-Maclaurin formula defined as follows:
With the Euler-Maclaurin formula, Moak [6] obtained the following q-analogue of Stirling formula
In this paper, the Euler-Maclaurin formula is exploited to provide an expression for the q-factorial function as an infinite integral.
In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals and sums.
In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler-Maclaurin formula is
(15), however, the Euler-Maclaurin formula may be used to obtain a more easily evaluated, asymptotic expression for Q(A):
Also, it introduces the new q-operator to reach the q-analogue of Euler-Maclaurin formula.
The Euler-MacLaurin formula. Let m [greater than or equal to] 0, n [greater than or equal to] 1, and define h = (b - a) /n, [x.sub.j] = a + jh for j = 0, 1, ..., n.