Calculating Highly Oscillatory Integrals by Quadrature Methods

Highly oscillatory integrals of the form $I(f)=\int_{0}^{\infty} dx f(x) e^{i \omega g(x)}$ arise in various problems in dynamics, image analysis, optics, and other fields of physics and mathematics. Conventional approximation methods for such highly oscillatory integrals tend to give huge errors as frequency ($\omega$) $\rightarrow \infty$. Over years, various attempts have been made to get over this flaw by considering alternative quadrature methods for integration. One such method was developed by Filon in 1928, which Iserles {\it et al.\ }have recently extended. Using this method, Iserles {\it et al.\ }show that as $\omega \rightarrow \infty$, the error decreases further as the error is inversely proportional to $\omega$. We use methods developed by Iserles' group, along with others like Newton-Cotes, Clenshaw-Curtis and Levin's methods with the aid of {\it Mathematica}. Our aim is to find a systematic way of calculating highly oscillatory integrals. In particular, our focus is on the oscillatory integrals that came up in earlier study of vacuum energy by Dr. Stephen Fulling