Asymptotic expansion and quadrature of composite highly oscillatory integrals

We consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form ∫ b a f(x)g(sin[ωθ(x)])dx, where ω 1. Such integrals occur in the simulation of electronic circuits, but they are also of independent mathematical interest. The integral is expanded in asymptotic series in inverse powers of ω. This expansion clarifies the behaviour for large ω and also provides a powerful means to design effective computational algorithms. In particular, we introduce and analyse Filon-type methods for this integral.