$I(x) = \int_a^b f(t) e^{ix\psi(t)} dt$   as $x\rightarrow \infty$

with $\psi(t)$ real.

Uses Riemann-Lebesgue lemma:

Riemann-Lebesgue lemma

... Useful also when doing integration by parts for Asymptotic approximation of integrals

See statement on notes

Method of stationary phase

Split integral into region close to stationary phase point(s) and the rest. Then it's similar to Laplace method

See example in notes..

Important notes

• The error terms are only algebraically small, not exponentially small as in Laplace method.
• Higher-order corrections are very hard to get since they may come from the whole range of integration.This is in contrast to Laplace method where the full asymptotic expansion depends only on the local region because the errors are exponentially small.