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

Abstract

The Euler-MacLauren formula

$$\int\limits_0^a {f(x)dx} = {I_t}(h) - \sum\limits_{i = 1}^m {{A_i}{h^{2i}} + {R_m}{h^{2m}}} $$

((1a))

where

$${A_i} = \frac{{{B_{2i}}}}{{2i!}}\left[ {{f^{(2i - 1)}}(nh) - {f^{(2i - 1)}}(0)} \right];\quad h = \frac{a}{n};$$

((1b))

B_n are the Bernoulli numbers, and I_t(h) is the trapezoidal sum:

$${R_m} = \frac{1}{{2m!}}{B_{2m}}{f^{(2m)}}(\xi ),\quad 0 \le \xi \le nh; $$

((1c))

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.