Numerical evaluation of a fixed-amplitude variable-phase integral

Abstract

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.