# Euler–Maclaurin Summation Formula: Definition

## What is the Maclaurin Summation Formula?

The Euler–Maclaurin summation formula, discovered independently by Leonhard Euler (in 1732) and Colin Maclaurin (in 1742), relates the summation of a function to an integral approximation. It gives a way to calculate “corrections” in terms of the function’s derivatives, evaluated at the endpoints. It’s most general form is : Where:

In addition, the function f must have a continuous kth derivative.

If the function and all of its derivatives approach zero as x→∞, the Euler–Maclaurin summation Formula simplifies (by letting b→∞ in the identity) to: It can also be expressed in a more “modern” asymmetrical form as : With a remainder, at any stage, of and The Euler-Maclaurin summation formula has many uses in mathematics and science, including quantum topology, where the formula is sometimes needed for a resurgent function with singularities in the vertical strip perpendicular to the range of summation . It is also one way to generate an asymptotic series.

## What are Bernoulli Numbers?

Bernoulli numbers are rational numbers defined as coefficients in the series expansion These coefficients are very challenging to calculate by hand; software is normally used to find them .
Bernoulli numbers Bk are defined as values of the Bernoulli polynomials Bk(t) at t = 0: Bk = Bk(0). Bernoulli polynomials satisfy the differential equation 
B′k+1(t) = (k + 1)Bk(t), with B2k+1(0) = 0, B2k+1(1) = 0, for all k > 0.

## References

 Kac, V. (2005). 18.704 Seminar in Algebra and Number Theory Fall 2005. Euler-Maclaurin Formula. Retrieved August 14, 2021 from: http://people.csail.mit.edu/kuat/courses/euler-maclaurin.pdf
Mills, S. (1985). The Independent Derivations by Leonhard Euler and Colin MacLaurin of the Euler-MacLaurin Summation Formula. Archive for History of Exact Sciences
Vol. 33, No. 1/3 (1985), pp. 1-13 (13 pages). Springer.
 Costin, O. & Garoufaldis, S. Resurgence of the E-M Summation Formula. Retrieved August 14, 2021 from: https://people.math.osu.edu/costin.9/EULMCL.pdf
 Rozman, M. (2016). Euler-Maclaurin summation formula. Retrieved August 14, 2021 from: https://www.phys.uconn.edu/~rozman/Courses/P2400_16S/downloads/euler-maclaurin-summation.pdf
 Verschelde, J. The Euler-Maclaurin Summation Formula. Retrieved August 14, 2021 from:
http://homepages.math.uic.edu/~jan/MCS471/Lec28/emsum.pdf

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