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

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 𝑓∈𝐶^2𝑚+2[𝑎,𝑏] the Euler-Maclaurin formula can be expressed as follows^𝑏_𝑎𝑓(𝑥)1𝑑𝑥=2𝑓(𝑎)+𝑓(𝑏)−^𝑚_𝑖=1𝐵_2𝑖(𝑏−𝑎)^2𝑖()!𝑓2𝑖^(2𝑖−1)(𝑏)−𝑓^(2𝑖−1)(𝑎)+𝑅_2𝑚+2, (1.1) where𝑅_2𝑚+2𝐵=−^2𝑚+2𝑓^(2𝑚+2)(𝜉)()!()2𝑚+2𝑏−𝑎^2𝑚+3, (1.2) and 𝜉 is some point between 𝑎 and 𝑏. The constants 𝐵_𝑖 are known as Bernoulli numbers, which are defined by the equation𝑥𝑒^𝑥=−1^∞_𝑘=0𝐵_𝑘𝑥𝑘!^𝑘. (1.3) The first few of the Bernoulli numbers are 𝐵₀=1, 𝐵₁=−1/2, 𝐵₂=1/6, and 𝐵_2𝑖−1=0 for all 𝑖≥2.

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 𝑎 and 𝑏 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 [1–6]. A direct generalization of the Euler-Maclaurin formula in the interval [𝑎,𝑎+ℎ] can be described as_𝑎^𝑎+ℎ𝑓(𝑥)ℎ𝑑𝑥=𝑓(𝑎)(+ℎ2𝑛+𝑓𝑎+ℎ)𝑛^𝑛−1_𝑘=1𝑓𝑘𝑎+𝑛ℎ−^𝑚_𝑖=1𝐵_2𝑖()!ℎ2𝑖𝑛^2𝑖𝑓^(2𝑖−1)(𝑎+ℎ)−𝑓^(2𝑖−1)(𝑎−𝑛𝐵)^2𝑚+2𝑓^(2𝑚+2)(𝜉)()!ℎ2𝑚+2𝑛^2𝑚+3, (1.4) where 𝜉∈(𝑎,𝑎+ℎ), 𝑛 is a positive integer and 𝑚 is a nonnegative integer. Obviously, when 𝑛=1, (1.4) reduces to (1.1). We also conclude that this equation has algebraic accuracy of 2𝑚+1 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 [7–14]. 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 polynomials𝐵_𝑛,𝑘=𝐵_𝑛,𝑘𝑥₁,𝑥₂,…,𝑥_{𝑛−𝑘+1} (2.1) in an infinite number of variables 𝑥₁,𝑥₂,…, defined by the series expansion1𝑘!_𝑚≥1𝑥_𝑚𝑡^𝑚𝑚!^𝑘=_𝑛≥𝑘𝐵_𝑛,𝑘𝑡^𝑛𝑛!,𝑘=0,1,2,…. (2.2) Their explicit expressions are given by the formula𝐵_𝑛,𝑘𝑥₁,𝑥₂,…,𝑥_{𝑛−𝑘+1}=𝑛!𝑎₁!(1!)^𝑎1𝑎₂!(2!)^𝑎2⋯𝑥₁^𝑎1𝑥₂^𝑎2…, (2.3) where the summation takes place over all nonnegative integers 𝑎₁,𝑎₂,…, such that 𝑎₁+2𝑎₂+⋯=𝑛 and 𝑎₁+𝑎₂+⋯=𝑘. For example, we have 𝐵_0,0=1,𝐵_1,1=𝑥₁,𝐵_2,1=𝑥₂,𝐵_2,2=𝑥₁²,𝐵_3,1=𝑥₃,𝐵_3,2=3𝑥₁𝑥₂,𝐵_3,3=𝑥₁³,…,𝐵_𝑛,1=𝑥_𝑛,𝐵_𝑛,𝑛=𝑥₁^𝑛. (2.4)

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 𝜉 in (1.4). Before the main result is given we first present an essential lemma.

Lemma 3.1 (see [15]). The following identity: ^𝑛_𝑘=1𝑘^𝑟=1𝑟+1^𝑟_𝑘=0𝐵_𝑘𝑘()𝑟+1𝑛+1^{𝑟+1−𝑘}, (3.1) holds, where 𝑟 is a positive integer.

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

Theorem 3.2. Let 𝑝, 𝑞 be integers and 𝑝≥1, 𝑞≥0. Assume that 𝑓 is a function admitting in a neighborhood of the point 𝑎 of ℝ a derivative of order 2𝑚+2+𝑝+𝑞 such that 𝑓^{(2𝑚+2+𝑝+𝑞)} is continuous at 𝑎. If 𝑓^(2𝑚+3)(𝑎)=⋯=𝑓^{(2𝑚+1+𝑝)}(𝑎)=0, and 𝑓^{(2𝑚+2+𝑝)}(𝑎)≠0, then 𝜉=𝑎+^𝑞+1_𝑖=1𝑐_𝑖ℎ^𝑖+𝑜ℎ^𝑞+1(ℎ⟶0). (3.2) The coefficients 𝑐_𝑖 are given by the recurrence formula 𝑐₁=(2𝑚+2)!𝑝!(2𝑚+3+𝑝)!𝐵^𝑝_{2𝑚+2𝑗=0}2𝑚+3+𝑝𝐵2𝑚+2+𝑗𝑛^{2𝑚+2+𝑗𝑗1/𝑝},𝑐_𝑖+1=𝑅_𝑖𝑐₁,…,𝑐_𝑖𝑖=1,…,𝑞 (3.3) with 𝑅_𝑖𝑐₁,…,𝑐_𝑖=1𝑡₀^1/𝑝𝑠₀^1/𝑝𝑖!^𝑖_𝑘=0𝑘1/𝑝𝑘!𝐵_𝑖,𝑘𝜙₁,𝜙₂,…,𝜙_{𝑖+1−𝑘}−𝑡₀^1/𝑝(𝑖+1)!^𝑖+1_𝑘=2𝑘𝐵_𝑖+1,𝑘1!𝑐₁,2!𝑐₂,…,(𝑖+2−𝑘)!𝑐_{𝑖+2−𝑘}^𝑘−1_𝑗=01𝑝𝑗𝑗!𝐵_{𝑘−1,𝑗}𝜓₁,𝜓₂,…,𝜓_𝑘−𝑗, (3.4) where 𝑠_𝑘=𝑓^{(2𝑚+2+𝑝+𝑘)}(𝑎)2𝑚+3+𝑝+𝑘!^𝑝+𝑘_𝑗=02𝑚+3+𝑝+𝑘𝐵2𝑚+2+𝑗𝑛^{2𝑚+2+𝑗𝑗},𝑡_𝑘=𝐵_2𝑚+2()!𝑓2𝑚+2^{(2𝑚+2+𝑝+𝑘)}(𝑎)!,𝜙𝑝+𝑘_𝑘=𝑠_𝑘𝑠₀,𝜓_𝑘=𝑡_𝑘𝑡₀,𝑘=0,1,…,𝑞. (3.5)

Proof. For convenience, we let 𝐴=_𝑎^𝑎+ℎ𝑓(𝑥)𝑛𝐵𝑑𝑥,𝐶=^2𝑚+2𝑓^(2𝑚+2)(𝜉)()!ℎ2𝑚+2𝑛^2𝑚+3,ℎ𝐵=𝑓(𝑎)(+ℎ2𝑛+𝑓𝑎+ℎ)𝑛^𝑛−1_𝑘=1𝑓𝑘𝑎+𝑛ℎ−^𝑚_𝑖=1𝐵_2𝑖()!ℎ2𝑖𝑛^2𝑖𝑓^(2𝑖−1)(𝑎+ℎ)−𝑓^(2𝑖−1)(𝑎). (3.6) According to (1.4), we have 𝐴=𝐵−C. We first consider 𝐴. Using the Taylor expansion, we have 𝐴=^{2𝑚+2+𝑝+𝑞}_𝑘=0𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑘+1+𝑜ℎ^{2𝑚+3+𝑝+𝑞},ℎ⟶0. (3.7) When ℎ→0, using the Taylor expansion again we have (𝑎)+1𝐵=ℎ𝑓2𝑛^{2𝑚+2+𝑝+𝑞}_𝑘=1𝑓^(𝑘)(𝑎)ℎ𝑘!^𝑘+1+ℎ𝑛^𝑛−1_𝑘=1^{2𝑚+2+𝑝+𝑞}_𝑗=1𝑓^(𝑗)(𝑎)𝑘𝑗!𝑛ℎ^𝑗−^𝑚_𝑘=1𝐵_2𝑘()!ℎ2𝑘𝑛^{2𝑘2𝑚+2+𝑝+𝑞}_𝑗=2𝑘𝑓^(𝑗)(𝑎)!ℎ𝑗+1−2𝑘^𝑗+1+𝑜ℎ^{2𝑚+3+𝑝+𝑞}(𝑎)+𝑓=ℎ𝑓^′(𝑎)ℎ2!²+12𝑛^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑓^(𝑘)(𝑎)ℎ𝑘!^𝑘+1+^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑓^(𝑘)(𝑎)ℎ𝑘!𝑛^{𝑘+1𝑛−1}_𝑗=1𝑗^𝑘−^2𝑚+1_𝑘=2𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^{⌊⌋𝑘+1𝑘/2}_𝑗=1𝑘+1𝐵2𝑗𝑛^2𝑗2𝑗−^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑚+2𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑚𝑘+1_𝑗=1𝑘+1𝐵2𝑗𝑛^2𝑗2𝑗+𝑜ℎ^{2𝑚+3+𝑝+𝑞}, (3.8) where ⌊𝑘/2⌋ denotes the largest integer that is not greater than 𝑘/2. Since 𝐵₀=1, 𝐵₁=−1/2, and 𝐵_2𝑖−1=0 for 𝑖≥2, by Lemma 3.1 there holds (𝑎)+𝑓𝐵=ℎ𝑓^′(𝑎)ℎ2!²+12𝑛^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑓^(𝑘)(𝑎)ℎ𝑘!^𝑘+1+^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑘𝑘+1_𝑗=0𝑗𝑘+1𝐵_𝑗𝑛^𝑗−^2𝑚+1_𝑘=2𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^{⌊⌋𝑘+1𝑘/2}_𝑗=1𝑘+1𝐵2𝑗𝑛^2𝑗2𝑗−^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑚+2𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑚𝑘+1_𝑗=1𝑘+1𝐵2𝑗𝑛^2𝑗2𝑗+𝑜ℎ^{2𝑚+3+𝑝+𝑞}=^2𝑚+1_𝑘=0𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑘+1+^{2𝑚+2+𝑝+𝑞}_𝑘=2𝑚+2𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑘+11+^𝑘_𝑗=2𝑚+2𝑗𝑘+1𝐵_𝑗𝑛^𝑗+𝑜ℎ^{2𝑚+3+𝑝+𝑞}, (3.9) as ℎ→0. Because 𝑓^(2𝑚+3)(𝑎)=⋯=𝑓^{(2𝑚+1+𝑝)}(𝑎)=0, we obtain 𝐵𝐵−𝐴=^2𝑚+2𝑓^(2𝑚+2)(𝑎)(2𝑚+2)!𝑛^2𝑚+2ℎ^2𝑚+3+^{2𝑚+2+𝑝+𝑞}_{𝑘=2𝑚+2+𝑝}𝑓^(𝑘)(𝑎)()!ℎ𝑘+1^𝑘𝑘+1_𝑗=2𝑚+2𝑗𝑘+1𝐵_𝑗𝑛^𝑗+𝑜ℎ^{2𝑚+3+𝑝+𝑞}, (3.10) as ℎ→0. By the Taylor expansion, it follows that 𝐵𝐶=^2𝑚+2ℎ^2𝑚+3(2𝑚+2)!𝑛^2𝑚+2𝑓^(2𝑚+2)(𝑎)+^𝑝+𝑞_𝑘=𝑝𝑓^{(2𝑚+2+𝑘)}(𝑎)()𝑘!𝜉−𝑎^𝑘+𝑜ℎ^𝑝+𝑞,ℎ⟶0. (3.11) Since 𝐵−𝐴=𝐶 and 0<𝜉−𝑎<ℎ, we have 𝜉−𝑎ℎ^𝑝𝐵_2𝑚+2(2𝑚+2)!^𝑞_𝑘=0𝑓^{(2𝑚+2+𝑝+𝑘)}(𝑎)!()𝑝+𝑘𝜉−𝑎^𝑘=^𝑞_𝑘=0𝑓^{(2𝑚+2+𝑝+𝑘)}(𝑎)!ℎ2𝑚+3+𝑝+𝑘^{𝑘𝑝+𝑘}_𝑗=02𝑚+3+𝑝+𝑘𝐵2𝑚+2+𝑗𝑛^{2𝑚+2+𝑗𝑗}+𝑜ℎ^𝑞,ℎ⟶0. (3.12) Let 𝑠_𝑘=𝑓^{(2𝑚+2+𝑝+𝑘)}(𝑎)2𝑚+3+𝑝+𝑘!^𝑝+𝑘_𝑗=02𝑚+3+𝑝+𝑘𝐵2𝑚+2+𝑗𝑛^{2𝑚+2+𝑗𝑗},𝑡_𝑘=𝐵_2𝑚+2()!𝑓2𝑚+2^{(2𝑚+2+𝑝+𝑘)}(𝑎)!,𝜙𝑝+𝑘_𝑘=𝑠_𝑘𝑠₀,𝜓_𝑘=𝑡_𝑘𝑡₀,𝑘=0,1,…,𝑞. (3.13) Then (3.12) can be rewritten as 𝜉−𝑎ℎ^𝑝𝑡₀^𝑞_𝑘=0𝜓_𝑘(𝜉−𝑎)^𝑘=𝑠₀^𝑞_𝑘=0𝜙_𝑘ℎ^𝑘+𝑜ℎ^𝑞,ℎ⟶0. (3.14) The rest proof is similar to that in [7, 8], and we omit it.

Putting 𝑚=0 in (1.4) we derive the composite trapezoidal rule as follows_𝑎^𝑎+ℎ𝑓(𝑥)ℎ𝑑𝑥=𝑓(𝑎)(+ℎ2𝑛+𝑓𝑎+ℎ)𝑛^𝑛−1_𝑘=1𝑓𝑘𝑎+𝑛ℎ−𝑓^′′(𝜉)12𝑛²ℎ³, (3.15) where 𝜉∈(𝑎,𝑎+ℎ). When ℎ→0, in view of Theorem 3.2 we obtain asymptotic behavior of the intermediate point 𝜉 in above composite trapezoidal rule.

Corollary 3.3. Let 𝑝, 𝑞 be integers and 𝑝≥1, 𝑞≥0. Assume that 𝑓 is a function admitting in a neighborhood of the point 𝑎 of ℝ a derivative of order 𝑝+𝑞+2 and that 𝑓^{(𝑝+𝑞+2)} is continuous at 𝑎. If 𝑓^′′′(𝑎)=⋯=𝑓^(𝑝+1)(𝑎)=0, and 𝑓^(𝑝+2)(𝑎)≠0, then 𝜉=𝑎+^𝑞+1_𝑖=1𝑐_𝑖ℎ^𝑖+𝑜ℎ^𝑞+1(ℎ⟶0). (3.16) The coefficients 𝑐_𝑖 are given by the recurrence formula 𝑐₁=13𝑝!(𝑝+3)!^𝑝_𝑗=0𝑝+3𝐵𝑗+2𝑛^{𝑗+2𝑗1/𝑝},𝑐_𝑖+1=𝑅_𝑖𝑐₁,…,𝑐_𝑖𝑖=1,…,𝑞 (3.17) with 𝑅_𝑖𝑐₁,…,𝑐_𝑖=1𝑡₀^1/𝑝𝑠₀^1/𝑝𝑖!^𝑖_𝑘=0𝑘1/𝑝𝑘!𝐵_𝑖,𝑘𝜙₁,𝜙₂,…,𝜙_{𝑖+1−𝑘}−𝑡₀^1/𝑝()!×𝑖+1_𝑖+1𝑘=2𝑘𝐵_𝑖+1,𝑘1!𝑐₁,2!𝑐₂,…,(𝑖+2−𝑘)!𝑐_{𝑖+2−𝑘}^𝑘−1_𝑗=0𝑗1/𝑝𝑗!𝐵_{𝑘−1,𝑗}𝜓₁,𝜓₂,…,𝜓_𝑘−𝑗, (3.18) where 𝑠_𝑘=𝑓^{(𝑝+𝑘+2)}(𝑎)𝑝+𝑘+3!^𝑝+𝑘_𝑗=0𝑝+𝑘+3𝐵𝑗+2𝑛^𝑗+2𝑗,𝑡_𝑘=1𝑓12^{(𝑝+𝑘+2)}(𝑎)!,𝜙𝑝+𝑘_𝑘=𝑠_𝑘𝑠₀,𝜓_𝑘=𝑡_𝑘𝑡₀,𝑘=0,1,…,𝑞. (3.19)

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).