---
tag: 119J1Q1MG9
kind: statement
role: theorem
name: Stationary phase approximation
msc: [41A60, 42B20]
uses: [0V33BNKZ9S, 0YTCMFB4HC, D5HZKE8E1A]
about: [D5HZKE8E1A]
sources:
  - url: https://en.wikipedia.org/wiki/Stationary_phase_approximation
same_as:
  wikidata: Q1936897
  wikipedia: Stationary phase approximation
---

## Statement

Let $g \colon \R \to \C$ be a [[smooth function|0V33BNKZ9S]] with compact support, and let $f \colon \R \to \R$ be smooth.
Assume that on the support of $g$ the function $f$ has exactly one [[critical point|0YTCMFB4HC]] $x_0$, and that $f''(x_0) \ne 0$.
Then, as $k \to +\infty$,
$$\int_{\R} g(x)\, e^{i k f(x)}\,dx = \sqrt{\frac{2\pi}{k \abs{f''(x_0)}}}\; g(x_0)\, e^{i k f(x_0) + i \frac{\pi}{4} \sgn f''(x_0)} + O\big(k^{-3/2}\big).$$

## 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 $k^{-1/3}$.

## Explanations

### L1 human

When $k$ is large, the factor $e^{ikf(x)}$ spins around the unit circle very fast as $x$ moves.
The positive and negative parts then cancel almost completely.
The cancellation fails only near points where the phase $f$ is stationary, that is, where $f'(x) = 0$.
So the whole integral is decided by a small neighbourhood of these points, and the theorem gives the exact leading term.

### L2 human

The stationary phase approximation is the oscillatory counterpart of [[Laplace's method|9G7K8DBT9Y]], which treats real exponents $e^{k f}$.
Both are special cases of the [[method of steepest descent|Q5H9VYFK98]] 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|CZCMRDWDYW]] and Airy functions and the decay of Fourier transforms of surface measures.
In physics it describes the passage from wave optics to geometric optics.