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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References
Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)
Brauchart, J.S., Hardin, D.P., Saff, E.B.: The Riesz energy of the Nth roots of unity: an asymptotic expansion for large N. Bull. Lond. Math. Soc. 41, 621–633 (2009)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
Ivić, A.: The Riemann Zeta-Function. Wiley, New York (1985)
Lyness, J.N.: Finite-part integrals and the Euler–Maclaurin expansion. In: Zahar, R.V.M. (ed.) Approximation and Computation. ISNM, vol. 119, pp. 397–407. Birkhäuser, Boston (1994)
Lyness, J.N., Ninham, B.W.: Numerical quadrature and asymptotic expansions. Math. Comput. 21, 162–178 (1967)
Monegato, G., Lyness, J.N.: The Euler–Maclaurin expansion and finite-part integrals. Numer. Math. 81, 273–291 (1998)
Navot, I.: An extension of the Euler–Maclaurin summation formula to functions with a branch singularity. J. Math. Phys. 40, 271–276 (1961)
Navot, I.: A further extension of the Euler–Maclaurin summation formula. J. Math. Phys. 41, 155–163 (1962)
Ninham, B.W.: Generalised functions and divergent integrals. Numer. Math. 8, 444–457 (1966)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis, 2nd edn. McGraw-Hill, New York (1978)
Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, vol. 10. Cambridge University Press, Cambridge (2003)
Sidi, A.: Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective. Numer. Math. 98, 371–387 (2004)
Sidi, A.: Extension of a class of periodizing variable transformations for numerical integration. Math. Comput. 75, 327–343 (2006)
Sidi, A.: A novel class of symmetric and nonsymmetric periodizing variable transformations for numerical integration. J. Sci. Comput. 31, 391–417 (2007)
Sidi, A.: Further extension of a class of periodizing variable transformations for numerical integration. J. Comput. Appl. Math. 221, 132–149 (2008)
Sidi, A.: Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comput. (in press)
Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3, 201–231 (1988). Originally appeared as Technical Report No. 384, Computer Science Dept., Technion–Israel Institute of Technology (1985), and also as ICASE Report No. 86–50 (1986)
Steffensen, J.F.: Interpolation, 2nd edn. Dover, New York (2006)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)
Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, New York (1986). Revised by D.R. Heath-Brown
Verlinden, P.: Cubature formulas and asymptotic expansions. PhD thesis, Katholieke Universiteit Leuven (1993). Supervised by A. Haegemans
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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