Euler-Maclaurin Integration Formulas
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
9-12-2021
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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
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
![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)],](https://mathworld.wolfram.com/images/equations/Euler-MaclaurinIntegrationFormulas/NumberedEquation4.gif) |
(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 
, 
, ..., 
:
![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))].](https://mathworld.wolfram.com/images/equations/Euler-MaclaurinIntegrationFormulas/NumberedEquation6.gif) |
(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.
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