Euler-Maclaurin formula
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In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. 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. 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.
If n is a natural number and f(x) is a smooth (meaning: sufficiently often differentiable) function defined for all real numbers x between 0 and n, then the integral
- <math>I=\int_0^n f(x)\,dx<math>
can be approximated by the sum
- <math>
S=\frac{f\left( 0\right) }{2}+f\left( 1\right) +\cdots+f\left( n-1\right) + \frac{f\left( n\right) }{2} <math> We can use two expressions for <math>S<math> :
- <math>S=-\frac{f\left( 0\right) +f\left( n\right) }{2}+\sum_{k=0}^{n}f\left(
n\right) <math> or
- <math>S=\frac{f\left( 0\right) +f\left( n\right) }{2}+\sum_{k=1}^{n-1}f\left(
n\right) <math>
(see trapezoidal rule). The Euler-Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives f(k) at the end points of the interval 0 and n. For any natural number p, we have
- <math>S-I=\sum_{k=1}^p\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(n)-f^{(2k-1)}(0)\right)+R<math>
where, B2 = 1/6, B4 = −1/30, B6 = 1/42, B8 = −1/30, ... are the Bernoulli numbers.
R is an error term which is normally small if p is large enough and can be estimated as
- <math>\left|R\right|\leq\frac{2}{(2\pi)^{2p}}\int_0^n\left|f^{(2p+1)}(x)\right|\,dx.<math>
By employing the substitution rule, one can adapt this formula also to functions f which are defined on some other interval of the real line.
If f is a polynomial and p is big enough, then the remainder term vanishes. For instance, if f(x) = x3, we can choose p = 2 to obtain after simplification
- <math>\sum_{i=0}^n i^3=\left(\frac{n(n+1)}{2}\right)^2.<math>
With the function f(x) = log(x), the Euler-Maclaurin formula can be used to derive precise error estimates for Stirling's approximation of the factorial function.
External links
- Bernoulli numbers, polynomials and applications of the Euler-Maclaurin formula (http://numbers.computation.free.fr/Constants/Miscellaneous/bernoulli.html)it:Formula di Euler - Maclaurin