The Euler-MacLauren Formula as an Asymptotic Form of Poisson’s

  • Henry E. Fettis
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)


The Euler-MacLauren formula
$$\int\limits_0^a {f(x)dx} = {I_t}(h) - \sum\limits_{i = 1}^m {{A_i}{h^{2i}} + {R_m}{h^{2m}}} $$
$${A_i} = \frac{{{B_{2i}}}}{{2i!}}\left[ {{f^{(2i - 1)}}(nh) - {f^{(2i - 1)}}(0)} \right];\quad h = \frac{a}{n};$$
Bn are the Bernoulli numbers, and It(h) is the trapezoidal sum:
$${R_m} = \frac{1}{{2m!}}{B_{2m}}{f^{(2m)}}(\xi ),\quad 0 \le \xi \le nh; $$
can be found in nearly every text on numerical analysis (e.g., [1]-[5]). Its importance lies in the fact that it is the basis for the derivation of quadrature formulae of Newton-Cotes type, i.e., quadrature formulae which utilize values of the integrand at equally spaced values of “x”. As is well known, this formula is not convergent, but asymptotic, in the sense that, whereas
$$ {h^{2m}}{R_m}(h) \to 0$$
for h → O (m fixed), it does not hold that
$${h^{2m}}{R_m}(h) \to 0$$
for m → ∞ (h fixed). Its conventional derivation is long and involved, and the reader is frequently at a loss to know how it could have originally been arrived at.


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    Cohen, A.M., Numerical Analysis, Halsted, N.Y., 1973Google Scholar
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    Davis, P.J. and Rabinowitz, P., Numerical Integration, Blaisdel, Waltham, Mass., 1967Google Scholar
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    Stoer, J. and Bulirsch, Introduction to Numerical Analysis, Springer, New York & Berlin, 1979Google Scholar
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    Stroud, A.H., Numerical Quadrature and Solution of Ordinary Differential Equations, Springer, New York & Berlin, 1974CrossRefGoogle Scholar
  6. 6.
    Hardy, G. H., Divergent series, Oxford, 1949, pp. 330–331Google Scholar

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© Springer Basel AG 1982

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  • Henry E. Fettis

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