Euler-Maclaurin summation formula
Let Br be the rth Bernoulli number, and Br(x) be the rth Bernoulli periodic function. For any integer k≥0 and for any function f of class Ck+1 on [a,b],a,b∈ℤ, we have
∑a<n≤bf(n)=∫baf(t)𝑑t+k∑r=0(-1)r+1Br+1(r+1)!(f(r)(b)-f(r)(a))+(-1)k(k+1)!∫baBk+1(t)f(k+1)(t)𝑑t. |
Title | Euler-Maclaurin summation formula |
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Canonical name | EulerMaclaurinSummationFormula |
Date of creation | 2013-03-22 11:46:01 |
Last modified on | 2013-03-22 11:46:01 |
Owner | KimJ (5) |
Last modified by | KimJ (5) |
Numerical id | 9 |
Author | KimJ (5) |
Entry type | Theorem |
Classification | msc 65B15 |
Classification | msc 00-02 |
Related topic | BernoulliNumber |