method of stationary phase

When you sum up many waves whose phases are spinning at different rates, most of them cancel each other into a blur — except where the phase momentarily stops changing, and neighbouring waves briefly march in step and add up. The method of stationary phase turns this everyday fact about waves into a tool: an oscillatory integral is dominated by the points where its phase is stationary, and almost everything else cancels.

The setting is an integral of g(t) e^(i k phi(t)) dt with a large real parameter k. The factor e^(i k phi) spins round and round; over any stretch where phi'(t) is not zero, the rapid spinning makes positive and negative contributions cancel, leaving a piece that is smaller than any power of 1/k. The surviving contribution comes from stationary points where phi'(t0) = 0, because there the phase is locally flat and the oscillations briefly reinforce. Expanding phi to second order there and doing a (now complex) Gaussian integral gives the leading estimate: about g(t0) sqrt(2 pi / (k |phi''(t0)|)) times e^(i k phi(t0)) e^(i sigma pi/4), where sigma is the sign of phi''(t0). Notice the decay is only like 1/sqrt(k) — slower than Laplace's method, because here oscillation cancels rather than exponential smallness.

Stationary phase is the mathematics of waves and rays: it explains why light and sound travel along stationary-time paths (the link to Fermat's principle), it gives the far-field diffraction patterns of optics and acoustics, the group velocity and dispersion of wave packets, and the WKB connection formulae. The honest caveat: it requires genuine, well-separated, non-degenerate stationary points. If phi''(t0) also vanishes (a higher-order stationary point), the decay slows further and special functions like the Airy function enter; and if there are no stationary points in the range, the integral is exponentially small and dominated by the endpoints instead.