Numerical Approximation of Highly Oscillatory Integrals

In this thesis, we introduce new methods for numerically approximating the integral of highly oscillatory function. Oscillatory integrals have many application, including solving oscillatory di erential equtions. Such integrals have an unwarranted reputation for being di cult to compute. We will demonstrate that high oscillation is in fact beneifcial:the methods discussed improve with accuracy as the frequency of oscillations increases. The asymptotic expansion will provide a point of departure, allowing us to prove that other, convergent methods have the same asymptotic behaviour, up to arbitrarily high order. This includes Filon-type methods, which require moments

Keywords:

Oscillatory integral , Filon-Type quadrature , asymptotic quadrature