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  "body": "\n## Statement\n\nLet $g \\colon \\R \\to \\C$ be a [[smooth function|0V33BNKZ9S]] with compact support, and let $f \\colon \\R \\to \\R$ be smooth.\nAssume that on the support of $g$ the function $f$ has exactly one [[critical point|0YTCMFB4HC]] $x_0$, and that $f''(x_0) \\ne 0$.\nThen, as $k \\to +\\infty$,\n$$\\int_{\\R} g(x)\\, e^{i k f(x)}\\,dx = \\sqrt{\\frac{2\\pi}{k \\abs{f''(x_0)}}}\\; g(x_0)\\, e^{i k f(x_0) + i \\frac{\\pi}{4} \\sgn f''(x_0)} + O\\big(k^{-3/2}\\big).$$\n\n## Remarks\n\nProof 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 $N$.\nA partition of unity therefore reduces the problem to a small neighbourhood of $x_0$.\nThere 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.\n\nExample: 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 $\\int_{\\R} e^{i k x^2}\\,dx = \\sqrt{\\pi / k}\\, e^{i\\pi/4}$.\n\nThe 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 $k^{-1/3}$.\n\n## Explanations\n\n### L1 human\n\nWhen $k$ is large, the factor $e^{ikf(x)}$ spins around the unit circle very fast as $x$ moves.\nThe positive and negative parts then cancel almost completely.\nThe cancellation fails only near points where the phase $f$ is stationary, that is, where $f'(x) = 0$.\nSo the whole integral is decided by a small neighbourhood of these points, and the theorem gives the exact leading term.\n\n### L2 human\n\nThe stationary phase approximation is the oscillatory counterpart of [[Laplace's method|9G7K8DBT9Y]], which treats real exponents $e^{k f}$.\nBoth are special cases of the [[method of steepest descent|Q5H9VYFK98]] when the functions extend holomorphically.\nWith several nondegenerate critical points the contributions add up, and there is a full asymptotic expansion in powers of $k^{-1/2 - j}$.\nIn $\\R^n$ 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.\nIt gives the asymptotics of [[Bessel functions|CZCMRDWDYW]] and Airy functions and the decay of Fourier transforms of surface measures.\nIn physics it describes the passage from wave optics to geometric optics.\n",
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