数値積分の漸近展開によるEuler-Maclaurin総和公式の簡単な別証明

Abstract

The Euler-Maclaurin summation formula plays an important role when we yield the correction formulas of numerical integrations or when we make the Romberg integration list. But the proof of the above summation formula, given by using the properties of Bernoulli polynomial, is fairly complicated. In this paper, fist we obtain some asymptotic expansions of numerical integration formulas by the terms of the mesuration by parts of higher order derivatives. As an application, we derive so-called end-point correction formulas of mesuration by parts, mid-point rule, trapezoid rule and Simpson's rule independently from the Euler-Maclaurin summation formula. Finally, we shall give an annother simpler and fundamental proof of the Euler-Maclaurin summation formula itself.