BIT Numerical Mathematics

, Volume 49, Issue 2, pp 397–417 | Cite as

Asymptotic expansions for oscillatory integrals using inverse functions

Article

Abstract

We treat finite oscillatory integrals of the form a b F(x)e ikG(x) dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient a b φ(t)e ikt dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when \(\phi(t)=\sum a_{j}t^{\sigma_{j}}\) , −1<σ 1<σ 2<⋅⋅⋅.

Keywords

Variable phase oscillatory integral Inverse functions Series inversion Fourier coefficient asymptotic expansion Fourier integral 

Mathematics Subject Classification (2000)

65D30 42A16 41A60 30E15 30E20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables. Nat. Bur. Standards Appl. Math. Series, vol. 55. U.S. Govt. Printing Office, Washington (1964) MATHGoogle Scholar
  2. 2.
    Bürmann, H.H.: Mémoires de l’Institut National des Sci. et Acts: Sci. Math. Phys. 2, 13–17 (1799) Google Scholar
  3. 3.
    Copson, E.T.: Theory of Functions of a Complex Variable. Oxford University Press, London (1935) Google Scholar
  4. 4.
    de Moivre, A.: A method of raising an infinite multinomial to any given power, or extracting any given root of the same. Philos. Trans. 19, 619–625 (1697) Google Scholar
  5. 5.
    Erdelyi, A.: Asymptotic representations of Fourier integrals and the method of stationary phase. J. Soc. Ind. Appl. Math. 3, 17–27 (1955) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Filon, J.N.G.: On a quadrature formula for trigonometric integrals. Proc. R. Soc. Edinb. 49, 38–47 (1928) Google Scholar
  7. 7.
    Hurwitz, A., Courant, R.: Allgemeine Funktionentheorie. Springer, Berlin (1925) MATHGoogle Scholar
  8. 8.
    Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026–1048 (2006) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Huybrechs, D., Olver, S.: Highly oscillatory quadrature. In: Highly Oscillatory Problems: Computation, Theory and Application. Cambridge University Press, Cambridge (2009) Google Scholar
  10. 10.
    Iserles, A., Nørsett, S.P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT 44, 755–772 (2004) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109, 217–234 (2002) MATHCrossRefGoogle Scholar
  12. 12.
    Lagrange, G.L.: Mem. Acad. R. Sci. (Berlin) 24, 251 (1768) Google Scholar
  13. 13.
    Landau, E.: Sitz.ber. Preuss. Akad. Wiss. Berl. Philos. Hist. Kl. 1118–1133 (1904) Google Scholar
  14. 14.
    Landau, E.: Math. Z. 30, 616–617 (1929) Google Scholar
  15. 15.
    Levin, D.: Analysis of a collocation method for integrating rapidly oscillatory functions. J. Comput. Appl. Math. 78, 131–138 (1997) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lighthill, M.J.: Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, Cambridge (1958) Google Scholar
  17. 17.
    Lyness, J.N.: Adjusted forms of the Fourier coefficient asymptotic expansion and application in numerical quadrature. Math. Comput. 25(113), 87–104 (1971) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lyness, J.N.: Numerical evaluation of a fixed-amplitude variable-phase integral. Numer. Algorithms 49, 235–249 (2008) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, San Diego (1974) Google Scholar
  20. 20.
    Olver, S.: Moment-free numerical approximation of highly oscillatory integrals with stationary points. Eur. J. Appl. Math. 18, 435–447 (2007) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis. Cambridge University Press, Cambridge (1953) Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA
  2. 2.School of MathematicsUniversity of New South WalesSydneyAustralia
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations