Oscillatory Integrals

Consider the “Fourier-type” integral

and we want the Asymptotic Approximation as . Note that this is different to Laplace integrals because here the integrand oscillates like crazy instead of blowing up, leading to catastrophic cancellation

is the amplitude / envelope functions is the phase is the local wavenumber

Riemann-Lebesgue Lemma Oscillatory Integrals with monotonic phase The Fresnel Integrals The Method of Stationary Phase