Numerical Algorithms

, Volume 49, Issue 1, pp 235–249

Numerical evaluation of a fixed-amplitude variable-phase integral

Original Paper

DOI: 10.1007/s11075-008-9166-y

Cite this article as:
Lyness, J.N. Numer Algor (2008) 49: 235. doi:10.1007/s11075-008-9166-y


We treat the evaluation of a fixed-amplitude variable-phase integral of the form \(\int_a^b \exp[ikG(x)]dx \), where G′(x) ≥ 0 and has moderate differentiability in the integration interval. In particular, we treat in detail the case in which G′(a) = G′(b) = 0, but G′′(a)G′′(b) < 0. For this, we re-derive a standard asymptotic expansion in inverse half-integer inverse powers of k. This derivation is direct, making no explicit appeal to the theories of stationary phase or steepest descent. It provides straightforward expressions for the coefficients in the expansion in terms of derivatives of G at the end-points. Thus it can be used to evaluate the integrals numerically in cases where k is large. We indicate the generalizations to the theory required to cover cases where the oscillator function G has higher order zeros at either or both end-points, but this is not treated in detail. In the simpler case in which G′(a)G′(b) > 0, the same approach would recover a special case of a recent result due to Iserles and Nørsett.


Fixed-amplitude variable-phase integral Stationary phase asymptotic expansions Highly oscillatory integrands Series inversion Steepest descent integration 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

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

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