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 $\int_a^b f(x) g(\sin[\omega \theta(x)])\mathrm{d} x$, where $\omega\gg1$. 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 $\omega$. This expansion clarifies the behaviour for large $\omega$ and also provides a powerful means to design effective computational algorithms. In particular, we introduce and analyse Filon-type methods for this integral.