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Euler-Maclaurin Integration Formulas

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The Euler-Maclaurin integration and sums formulas can be derived from Darboux's formula by substituting the Bernoulli polynomial B_n(t) in for the function phi(t). Differentiating the identity

B_n(t+1)-B_n(t)=nt^(n-1)
(1)

n-k times gives

B_n^((n-k))(t+1)-B_n^((n-k))(t)=n(n-1)...kt^(k-1).
(2)

Plugging in t=0 gives B_n^((n-k))(1)=B_n^((n-k))(0). From the Maclaurin series of B_n(z) with k>0, we have

B_n^((n-2k-1))(0)=0
(3)
B_n^((n-2k))(0)=(n!)/((2k)!)B_(2k)
(4)
B_n^((n-1))(0)=1/2n!
(5)
B_n^((n))(0)=n!,
(6)

where B_n is a Bernoulli number, and substituting these values of B_n^((n-k))(1) and B_n^((n-k))(0) into Darboux's formula gives

(z-a)f^'(a)=f(z)-f(a)-(z-a)/2[f^'(z)-f^'(a)]+sum_(m=1)^(n-1)(B_(2m)(z-a)^(2m))/((2m)!)[f^((2m))(z)-f^((2m))(a)]-((z-a)^(2n+1))/((2n)!)int_0^1B_(2n)(t)f^((2n+1))[a-(z-a)t]dt,
(7)

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

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

sum_(k=1)^(n-1)f_k=int_0^nf(k)dk-1/2[f(0)+f(n)]+sum_(k=1)^infty(B_(2k))/((2k)!)[f^((2k-1))(n)-f^((2k-1))(0)],
(8)

which, when expanded, gives

sum_(k=1)^(n-1)f_k=int_0^nf(k)dk-1/2[f(0)+f(n)]+1/(12)[f^'(n)-f^'(0)]-1/(720)[f^(''')(n)-f^(''')(0)]+1/(30240)[f^((5))(n)-f^((5))(0)]-1/(1209600)[f^((7))(n)-f^((7))(0)]+...
(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 f(x) is tabulated at n values f_(3/2), f_(5/2), ..., f_(n-1/2):

int_(x_1)^(x_n)f(x)dx=h[f_(3/2)+f_(5/2)+f_(7/2)+...+f_(n-3/2)+f_(n-1/2)] -sum_(k=1)^infty(B_(2k)h^(2k))/((2k)!)(1-2^(-2k+1))[f_n^((2k-1))-f_1^((2k-1))].
(10)

See also

Darboux's Formula, Maclaurin-Cauchy Theorem, Sum, Wynn's Epsilon Method

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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.

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Euler-Maclaurin Integration Formulas

Cite this as:

Weisstein, Eric W. "Euler-Maclaurin Integration Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html

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