منابع مشابه
Euler - Maclaurin Formula
a Bk({1− t}) k! f (t)dt where a and b are arbitrary real numbers with difference b − a being a positive integer number, Bn and bn are Bernoulli polynomials and numbers, respectively, and k is any positive integer. The condition we impose on the real function f is that it should have continuous k-th derivative. The symbol {x} for a real number x denotes the fractional part of x. Proof of this th...
متن کاملThe Euler - Maclaurin Expansion
We have seen that the accuracy of methods for computing integrals or derivatives of a function f(x) depends on the spacing between points at which f is evaluated, and that the approximation tends to the exact value as this spacing tends to 0. Suppose that a uniform spacing h is used. We denote by F (h) the approximation computed using the spacing h, from which it follows that the exact value is...
متن کاملEuler-Maclaurin and Gregory interpolants
Let a sufficiently smooth function f on [−1, 1] be sampled at n + 1 equispaced points, and let k ≥ 0 be given. An Euler–Maclaurin interpolant to the data is defined, consisting of a sum of a degree k algebraic polynomial and a degree n trigonometric polynomial, which deviates from f by O(n−k) and whose integral is equal to the order k Euler–Maclaurin approximation of the integral of f . This in...
متن کاملEuler–Maclaurin summation and Schlömilch series
A method for analysing a class of divergent series is developed from the Euler– Maclaurin summation formula. The conditions that the summand must satisfy are explored, and a significant simplification is obtained for cases where the summation ranges over all integers. As an example, we consider the Ewald representation for Schlömilch series, and show that this includes Twersky’s dual series for...
متن کاملAsymptotic Euler-Maclaurin formula for Delzant polytopes
Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin-Sternberg [GS]. Then, the problem is to find a concrete formula for the each term of the e...
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ژورنال
عنوان ژورنال: Mathematical Inequalities & Applications
سال: 2003
ISSN: 1331-4343
DOI: 10.7153/mia-06-24