BIT Numerical Mathematics

, Volume 49, Issue 2, pp 397–417

Asymptotic expansions for oscillatory integrals using inverse functions

Article

DOI: 10.1007/s10543-009-0223-2

Cite this article as:
Lyness, J.N. & Lottes, J.W. Bit Numer Math (2009) 49: 397. doi:10.1007/s10543-009-0223-2

Abstract

We treat finite oscillatory integrals of the form abF(x)eikG(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 abφ(t)eiktdt. 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<σ1<σ2<⋅⋅⋅.

Keywords

Variable phase oscillatory integralInverse functionsSeries inversionFourier coefficient asymptotic expansionFourier integral

Mathematics Subject Classification (2000)

65D3042A1641A6030E1530E20

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