TOPICS
Search

Euler-Maclaurin Integration Formulas

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The Euler-Maclaurin integration formulas relate integrals to sums and endpoint derivatives. They follow from Darboux's formula by substituting a Bernoulli polynomial B_n(t) for 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!)/((k-1)!)t^(k-1)
(2)

for 1<=k<=n. Hence the endpoint values agree for 2<=k<=n, while

B_n^((n-1))(0)=-1/2n!
(3)
B_n^((n-1))(1)=1/2n!.
(4)

The remaining values needed in the substitution are

B_n^((n-2k-1))(0)=0
(5)
B_n^((n-2k))(0)=(n!)/((2k)!)B_(2k)
(6)
B_n^((n))(t)=n!,
(7)

where the first two formulas use positive integers k with 2k+1<=n and 2k<=n, respectively, and B_(2k) is a Bernoulli number.

For a positive integer n, the resulting even-order integration formula is

f(z)-f(a)=(z-a)/2[f^'(z)+f^'(a)]-sum_(m=1)^n(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.
(8)

This holds when f is analytic near the integration line segment. For a real interval, 2n+1 continuous derivatives suffice (Whittaker and Watson 1990, p. 128).

For positive integers N and p, the finite Euler-Maclaurin sum formula is

sum_(j=1)^(N-1)f(j)=int_0^Nf(x)dx-1/2[f(0)+f(N)]+sum_(k=1)^p(B_(2k))/((2k)!)[f^((2k-1))(N)-f^((2k-1))(0)]+R_p,
(9)

where

R_p=-1/((2p)!)int_0^NP_(2p)(x)f^((2p))(x)dx
(10)

and P_(2p)(x)=B_(2p)(x-|_x_|) is the periodic Bernoulli polynomial, with |_x_| denoting the floor function. This finite identity requires f to have 2p continuous derivatives on [0,N] (NIST, §2.10).

An infinite expansion follows if R_p->0 as p->infty. The first terms are

sum_(j=1)^(N-1)f(j)=int_0^Nf(x)dx-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)]+...
(11)

(Abramowitz and Stegun 1972, p. 16). Without a vanishing remainder, the infinite expansion need not converge. The Euler-Maclaurin sum formula is implemented in the Wolfram Language as the function NSum with option Method -> "EulerMaclaurin".

For the midpoint version, let x_j=x_1+(j-1)h, where h=(x_N-x_1)/(N-1)>0 and N>=2. Write f_j=f(x_j) and f_j^((r))=f^((r))(x_j). The N-1 midpoint samples are f_(3/2), f_(5/2), ..., f_(N-1/2). For a positive integer p,

int_(x_1)^(x_N)f(x)dx=hsum_(j=1)^(N-1)f_(j+1/2)+sum_(k=1)^p(B_(2k)h^(2k))/((2k)!)(1-2^(1-2k))[f_N^((2k-1))-f_1^((2k-1))]+R_p^((M)),
(12)

where

R_p^((M))=(h^(2p))/((2p)!)int_(x_1)^(x_N)P_(2p)((x-x_1)/h+1/2)f^((2p))(x)dx.
(13)

This holds when f has 2p continuous derivatives on [x_1,x_N]. The infinite midpoint expansion follows when this remainder tends to 0. Its correction sum has a positive sign. It can also be derived by writing the midpoint sum as 2T_(h/2)-T_h, where T_h is the composite trapezoidal rule with step h, and applying the finite Euler-Maclaurin formula to both terms.

See also

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

Explore with Wolfram|Alpha

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, Scotland, p. 672, 1742.National Institute of Standards and Technology. "Sums and Sequences." §2.10 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/2.10.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.

Referenced on Wolfram|Alpha

Euler-Maclaurin Integration Formulas

Cite this as:

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

Subject classifications