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Concept
Euler–Maclaurin formula
Summary
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.
The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula.
If m and n are natural numbers and f(x) is a real or complex valued continuous function for real numbers x in the interval [m,n], then the integral
can be approximated by the sum (or vice versa)
(see rectangle method). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives f^(k)(x) evaluated at the endpoints of the interval, that is to say x = m and x = n.
Explicitly, for p a positive integer and a function f(x) that is p times continuously differentiable on the interval [m,n], we have
where Bk is the kth Bernoulli number (with B1 = 1/2) and Rp is an error term which depends on n, m, p, and f and is usually small for suitable values of p.
The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for B1. In this case we have
or alternatively
The remainder term arises because the integral is usually not exactly equal to the sum. The formula may be derived by applying repeated integration by parts to successive intervals [r, r + 1] for r = m, m + 1, ..., n − 1. The boundary terms in these integrations lead to the main terms of the formula, and the leftover integrals form the remainder term.
The remainder term has an exact expression in terms of the periodized Bernoulli functions Pk(x).
Official source
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