Euler-Maclaurin Integration Formulas

The Euler-Maclaurin integration and sums formulas can be derived from Darboux's formula by substituting the Bernoulli polynomial  in for the function . Differentiating the identity

(1)

 times gives

(2)

Plugging in  gives . From the Maclaurin series of  with , we have

			(3)
			(4)
			(5)
			(6)

where  is a Bernoulli number, and substituting these values of  and  into Darboux's formula gives

(7)

which is the Euler-Maclaurin integration formula (Whittaker and Watson 1990, p. 128). It holds when the function  is analytic in the integration region

In certain cases, the last term tends to 0 as , and an infinite series can then be obtained for . In such cases, sums may be converted to integrals by inverting the formula to obtain the Euler-Maclaurin sum formula

(8)

which, when expanded, gives

(9)

(Abramowitz and Stegun 1972, p. 16). The Euler-Maclaurin sum formula is implemented in the Wolfram Language as the function NSum with option Method -> "EulerMaclaurin".

The second Euler-Maclaurin integration formula is used when  is tabulated at  values , , ..., :

(10)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 16 and 806, 1972.

Apostol, T. M. "An Elementary View of Euler's Summation Formula." Amer. Math. Monthly 106, 409-418, 1999.

Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula." §5.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 327-338, 1985.

Borwein, J. M.; Borwein, P. B.; and Dilcher, K. "Pi, Euler Numbers, and Asymptotic Expansions." Amer. Math. Monthly 96, 681-687, 1989.

Euler, L. Comm. Acad. Sci. Imp. Petrop. 6, 68, 1738.

Havil, J. "Euler-Maclaurin Summation." §10.2 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 85-86, 2003.

Knopp, K. Theory and Application of Infinite Series. New York: Dover, 1990.

Maclaurin, C. Treatise of Fluxions. Edinburgh, p. 672, 1742.

Vardi, I. "The Euler-Maclaurin Formula." §8.3 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 159-163, 1991.

Whittaker, E. T. and Robinson, G. "The Euler-Maclaurin Formula." §67 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 134-136, 1967.

Whittaker, E. T. and Watson, G. N. "The Euler-Maclaurin Expansion." §7.21 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 127-128, 1990.