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