Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2010 (2010), Article ID 134392, 8 pages
http://dx.doi.org/10.1155/2010/134392
Research Article

Asymptotic Behaviors of Intermediate Points in the Remainder of the Euler-Maclaurin Formula

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Received 23 June 2010; Revised 22 October 2010; Accepted 5 December 2010

Academic Editor: Dumitru Baleanu

Copyright © 2010 Aimin Xu and Zhongdi Cen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Euler-Maclaurin formula is a very useful tool in calculus and numerical analysis. This paper is devoted to asymptotic expansion of the intermediate points in the remainder of the generalized Euler-Maclaurin formula when the length of the integral interval tends to be zero. In the special case we also obtain asymptotic behavior of the intermediate point in the remainder of the composite trapezoidal rule.

1. Introduction

It is well known that the Euler-Maclaurin formula is a formula used in the numerical evaluation of integral, which states that the value of an integral is equal to the sum of the value given by the trapezoidal rule and a series of terms involving the odd-numbered derivatives of the function at the end points of the integral interval. Specifically, for the function the Euler-Maclaurin formula can be expressed as follows where and is some point between and . The constants are known as Bernoulli numbers, which are defined by the equation The first few of the Bernoulli numbers are , , , and for all .

The Euler-Maclaurin formula was discovered independently by Leonhard Euler and Colin Maclaurin, and it has wide applications in calculus and numerical analysis. For example, the Euler-Maclaurin formula is often used to evaluate finite sums and infinite series when and are integers. Conversely, it is also used to approximate integrals by finite sums. Therefore, the Euler-Maclaurin formula provides the correspondence between sums and integrals. Besides, the Euler-Maclaurin formula may be used to derive a wide range of quadrature formulas including the Newton-Cotes formulas, and used for detailed error analysis in numerical quadrature.

The Euler-Maclaurin formula has many generalizations and extensions [16]. A direct generalization of the Euler-Maclaurin formula in the interval can be described as where , is a positive integer and is a nonnegative integer. Obviously, when , (1.4) reduces to (1.1). We also conclude that this equation has algebraic accuracy of which is the same as (1.1).

Recently, some interests have been focused on the study of the mean value theorem for integrals and differentiations [714]. The aim of the present paper is to deal with asymptotic expansions of the intermediate points in the generalized Euler-Maclaurin formula when the length of the integral interval tends to be zero. The rest of this paper is organized as follows. In the second section, the Bell polynomials as a standard mathematical are introduced in detail. In the third section, we give asymptotic behavior of the intermediate points in the remainder of the generalized Euler-Maclaurin formula (1.4). As a special case, asymptotic behavior of the intermediate points in remainder of the composite trapezoidal formula is also presented.

2. Bell Polynomials

The Bell polynomials [15] extensively studied by Bell arise in combinatorial analysis, and they have been applied in many different frameworks. The exponential partial Bell polynomials are the polynomials in an infinite number of variables , defined by the series expansion Their explicit expressions are given by the formula where the summation takes place over all nonnegative integers , such that and . For example, we have

For more important properties the reader is referred to [15].

3. Asymptotic Expansions of Intermediate Points

In this section, we will consider asymptotic behavior of the point in (1.4). Before the main result is given we first present an essential lemma.

Lemma 3.1 (see [15]). The following identity: holds, where is a positive integer.

This lemma gives relations between the sum of powers of the first integers and the Bernoulli numbers. Now, we turn to the asymptotic behavior of the point in (1.4). Namely, the following theorem is our main result.

Theorem 3.2. Let , be integers and , . Assume that is a function admitting in a neighborhood of the point of a derivative of order such that is continuous at . If , and , then The coefficients are given by the recurrence formula with where

Proof. For convenience, we let According to (1.4), we have . We first consider . Using the Taylor expansion, we have When , using the Taylor expansion again we have where denotes the largest integer that is not greater than . Since , , and for , by Lemma 3.1 there holds as . Because , we obtain as . By the Taylor expansion, it follows that Since and , we have Let Then (3.12) can be rewritten as The rest proof is similar to that in [7, 8], and we omit it.

Putting in (1.4) we derive the composite trapezoidal rule as follows where . When , in view of Theorem 3.2 we obtain asymptotic behavior of the intermediate point in above composite trapezoidal rule.

Corollary 3.3. Let , be integers and , . Assume that is a function admitting in a neighborhood of the point of a derivative of order and that is continuous at . If , and , then The coefficients are given by the recurrence formula with where

Acknowledgment

This work is supported by the Zhejiang Province Natural Science Foundation (Grant no. Y6100021), and the Ningbo Natural Science Foundation (Grant nos. 2010A610099 and 2009A610082).

References

  1. G. Rzadkowski and S. Łepkowski, “A generalization of the Euler-Maclaurin summation formula: an application to numerical computation of the Fermi-Dirac integrals,” Journal of Scientific Computing, vol. 35, no. 1, pp. 63–74, 2008. View at Publisher · View at Google Scholar
  2. J.-P. Berrut, “A circular interpretation of the Euler-Maclaurin formula,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 375–386, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. I. Franjić and J. Pečarić, “Corrected Euler-Maclaurin's formulae,” Rendiconti del Circolo Matematico di Palermo. Serie II, vol. 54, no. 2, pp. 259–272, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. Dedić, M. Matić, and J. Pečarić, “Euler-Maclaurin formulae,” Mathematical Inequalities & Applications, vol. 6, no. 2, pp. 247–275, 2003. View at Google Scholar · View at Zentralblatt MATH
  5. D. Elliott, “The Euler-Maclaurin formula revisited,” Journal of Australian Mathematical Society Series B, vol. 40, pp. E27–E76, 1998/99. View at Google Scholar
  6. F.-J. Sayas, “A generalized Euler-Maclaurin formula on triangles,” Journal of Computational and Applied Mathematics, vol. 93, no. 2, pp. 89–93, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. U. Abel, “On the Lagrange remainder of the Taylor formula,” American Mathematical Monthly, vol. 110, no. 7, pp. 627–633, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. U. Abel and M. Ivan, “The differential mean value of divided differences,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 560–570, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. B. Jacobson, “On the mean value theorem for integrals,” The American Mathematical Monthly, vol. 89, no. 5, pp. 300–301, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. B.-L. Zhang, “A note on the mean value theorem for integrals,” The American Mathematical Monthly, vol. 104, no. 6, pp. 561–562, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. W. J. Schwind, J. Ji, and D. E. Koditschek, “A physically motivated further note on the mean value theorem for integrals,” The American Mathematical Monthly, vol. 106, no. 6, pp. 559–564, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. R. C. Powers, T. Riedel, and P. K. Sahoo, “Limit properties of differential mean values,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 216–226, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. T. Trif, “Asymptotic behavior of intermediate points in certain mean value theorems,” Journal of Mathematical Inequalities, vol. 2, no. 2, pp. 151–161, 2008. View at Google Scholar · View at Zentralblatt MATH
  14. A. M. Xu, F. Cui, and H. Z. Chen, “Asymptotic behavior of intermediate points in the differential mean value theorem of divided differences with repetitions,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 358–362, 2010. View at Publisher · View at Google Scholar
  15. L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, The Netherlands, 1974.