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Euler–Maclaurin Summation Formula: Definition

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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 [1]:
euler maclaurin formula

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:
euler-maclaurin summation formula simplified 2

It can also be expressed in a more “modern” asymmetrical form as [2]:
asymmetrical form

With a remainder, at any stage, of
euler-maclaurin remainder

and
remainder 2

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 [3]. 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
bernoulli numbers

These coefficients are very challenging to calculate by hand; software is normally used to find them [4].
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 [5]
B′k+1(t) = (k + 1)Bk(t), with B2k+1(0) = 0, B2k+1(1) = 0, for all k > 0.

References

[1] 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
[2]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.
[3] 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
[4] 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
[5] 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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