Numerische Mathematik

, Volume 98, Issue 2, pp 371–387 | Cite as

Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective

Article

Summary.

In this note, we provide a new perspective on Euler–Maclaurin expansions of (offset) trapezoidal rule approximations of the finite-range integrals I[f]=∫baf(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 Open image in new window where Ps(y) and Qs(y) are some polynomials in y. Here the γs and δs are complex in general and different from −1,−2,... . The results we obtain in this work generalize, and include as special cases, those pertaining to the known special cases in which f(x)=(xa)γ[ log (xa)]pga(x)=(bx)δ[log (bx)]qgb(x), where p and q are nonnegative integers and ga ∈ C[a,b) and gb ∈ C(a,b]. In addition, they have the pleasant feature that they are expressed in very simple terms based only on the asymptotic expansions of f(x) as xa+ and xb−. With h=(ba)/n, where n is a positive integer, and with Open image in new window one of these results reads, as h→0,Open image in new window where ζ(z) is the Riemann Zeta function.

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References

  1. 1.
    Apostol, T.M.: Mathematical Analysis. Addison–Wesley, London, 1957Google Scholar
  2. 2.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York, second edition, 1984Google Scholar
  3. 3.
    Lyness, J.N.: Finite-part integrals and the Euler–Maclaurin expansion. In R.V.M. Zahar, editor, Approximation and Computation, number 119 in ISNM, pages 397–407, Boston–Basel–Berlin, 1994. Birkhäuser VerlagGoogle Scholar
  4. 4.
    Lyness, J.N., Ninham, B.W.: Numerical quadrature and asymptotic expansions. Math. Comp. 21, 162–178 (1967)MATHGoogle Scholar
  5. 5.
    Monegato, G., Lyness, J.N.: The Euler–Maclaurin expansion and finite-part integrals. Numer. Math. 81, 273–291 (1998)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Navot, I.: An extension of the Euler–Maclaurin summation formula to functions with a branch singularity. J. Math. and Phys. 40, 271–276 (1961)MATHGoogle Scholar
  7. 7.
    Navot, I.: A further extension of the Euler–Maclaurin summation formula. J. Math. and Phys. 41, 155–163 (1962)MATHGoogle Scholar
  8. 8.
    Ninham, B.W.: Generalised functions and divergent integrals. Numer. Math. 8, 444–457 (1966)MATHGoogle Scholar
  9. 9.
    Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Number 10 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2003Google Scholar
  10. 10.
    Steffensen, J.F.: Interpolation. Chelsea, New York, 1950Google Scholar
  11. 11.
    Verlinden, P.: Cubature formulas and asymptotic expansions. PhD thesis, Katholieke Universiteit Leuven, 1993. Supervised by A. HaegemansGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Computer Science DepartmentTechnion - Israel Institute of TechnologyHaifaIsrael

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