Numerical evaluation of a fixed-amplitude variable-phase integral

We treat the evaluation of a fixed-amplitude variable-phase integral of the form b a 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. However, this derivation provides straightforward expressions for the coefficients in terms of derivatives of G at the end-points. Thus it can be used to evaluate the integrals 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, this approach recovers a special case of a recent result due to Iserles and N

• Type: Journal
• Year: 2008
• Where: NA
• Authors: J. N. Lyness
• Keywords: Fixed-amplitude Variable-phase, Integration Interval, NA 2008, Standard Asymptotic Expansion