Stationary phase approximation

Remarks

Proof idea: if $f' \ne 0$ on the support of $g$, repeated integration by parts with $e^{ikf} = \frac{1}{ikf'} \frac{d}{dx} e^{ikf}$ shows that the integral is $O(k^{-N})$ for every $N$. A partition of unity therefore reduces the problem to a small neighbourhood of $x_0$. There the Morse lemma gives a change of variable with $f(x) = f(x_0) + \frac12 f''(x_0)\, y^2$, and the Fresnel integral $\int_{\R} e^{i a y^2 / 2}\,dy = \sqrt{2\pi / \abs{a}}\, e^{i \frac{\pi}{4} \sgn a}$ produces the main term.

Example: for $f(x) = x^2$ and $g$ equal to $1$ near $0$, the main term is $\sqrt{\pi / k}\, e^{i\pi/4}$, which matches $\int_{\R} e^{i k x^2}\,dx = \sqrt{\pi / k}\, e^{i\pi/4}$.

The hypothesis $f''(x_0) \ne 0$ is needed for this rate: for $f(x) = x^3$ the critical point at $0$ is degenerate, and the integral decays only like $k^{-1/3}$.

Going deeper

For a student

The stationary phase approximation is the oscillatory counterpart of Laplace's method, which treats real exponents $e^{k f}$. Both are special cases of the method of steepest descent when the functions extend holomorphically. With several nondegenerate critical points the contributions add up, and there is a full asymptotic expansion in powers of $k^{-1/2 - j}$. In $\R^n$ the factor becomes $(2\pi/k)^{n/2} \abs{\det f''(x_0)}^{-1/2} e^{i \frac{\pi}{4} \operatorname{sgn} f''(x_0)}$, where $\operatorname{sgn}$ of the Hessian is its signature. It gives the asymptotics of Bessel functions and Airy functions and the decay of Fourier transforms of surface measures. In physics it describes the passage from wave optics to geometric optics.