Euler–Maclaurin Formula - The Formula

The Formula

If m and n are natural numbers and f(x) is a smooth (meaning: sufficiently often differentiable) function defined for all real numbers x in the interval, then the integral

can be approximated by the sum (or vice versa)

(see trapezoidal rule). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ(k) at the end points of the interval m and n. Explicitly, for any natural number p, we have

where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, … are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p and depends on n, m, p and f. (The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for B1.)

Note that

Hence, we may also write the formula as follows:

\sum_{i=m}^n f(i) = \int^n_m f(x)\,dx - B_1 \left(f(n) + f(m)\right) + \sum_{k=1}^p\frac{B_{2k}}{(2k)!}\left(f^{(2k - 1)'}(n) - f^{(2k - 1)'}(m)\right) + R

or even compacter and more elegant

\sum_{i=m}^n f(i) = \sum_{k=0}^p\frac{1}{k!}\left(B_k f^{(k - 1)'}(n) - B^\ast_k f^{(k - 1)'}(m)\right) + R

with the convention of, i.e. the -1-th derivation of f is the integral of the function. This presentation also emphasizes the notation of the two kinds of Bernoulli numbers, called the first and the second kind, which are of equal legitimacy, and in principle not to favor of each other. Here we denote with the Bernoulli number of the second kind (only because the historical reason of formation of this article) which differ from the first kind only for the index 1.

Read more about this topic:  Euler–Maclaurin Formula

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