Abstract
In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals \(I[f]=\int^{b}_{a}f(x)\,dx\), where f∈C ∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms
where \(\widehat{P}(y),P_{s}(y)\) and \(\widehat{Q}(y),Q_{s}(y)\) are polynomials in y. The γ s and δ s are distinct, complex in general, and different from −1. They also satisfy
The results we obtain in this work extend the results of a recent paper [A. Sidi, Numer. Math. 98:371–387, 2004], which pertain to the cases in which \(\widehat{P}(y)\equiv0\) and \(\widehat{Q}(y)\equiv0\). They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let \(D_{\omega}=\frac{d}{d\omega}\), h=(b−a)/n, where n is a positive integer, and define \(\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)\). Then with \(\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}\) and \(\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}\), one of these results reads
where ζ(z) is the Riemann Zeta function and σ i are Stieltjes constants defined via \(\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]\), i=0,1,… .
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Communicated by Edward B. Saff.
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Sidi, A. Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities. Constr Approx 36, 331–352 (2012). https://doi.org/10.1007/s00365-011-9140-0
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DOI: https://doi.org/10.1007/s00365-011-9140-0
Keywords
- Euler–Maclaurin expansions
- Asymptotic expansions
- Trapezoidal rule
- Endpoint singularities
- Algebraic singularities
- Logarithmic singularities
- Hadamard finite part
- Zeta function
- Stieltjes constants