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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nLet $\\phi \\colon \\R^n \\to \\R$ be a [[smooth function|0V33BNKZ9S]], the *phase*.\nLet $\\psi \\colon \\R^n \\to \\C$ be a smooth function with compact support, the *amplitude*.\nThe *oscillatory integral* with phase $\\phi$ and amplitude $\\psi$ is the function of the real parameter $\\lambda$ given by\n$$I(\\lambda) = \\int_{\\R^n} e^{i\\lambda\\phi(x)} \\psi(x) \\, dx.$$\nA point $x$ with $\\nabla\\phi(x) = 0$ is a *critical point*, or stationary point, of the phase.\n\n## Remarks\n\nExample: $n = 1$, $\\phi(x) = x$ and $\\psi$ smooth with compact support.\nThen $I(\\lambda) = \\int e^{i\\lambda x}\\psi(x) \\, dx$ is the Fourier transform of $\\psi$ at $-\\lambda/(2\\pi)$, and repeated integration by parts gives $\\abs{I(\\lambda)} \\le C_N \\abs{\\lambda}^{-N}$ for every $N$.\nMore generally, if $\\nabla\\phi \\ne 0$ on the support of $\\psi$, then $I(\\lambda)$ decays faster than any power of $\\lambda$.\n\nNon-example: $\\phi(x) = x$ and $\\psi \\equiv 1$ on $\\R$.\nThe amplitude does not have compact support, and $\\int_\\R e^{i\\lambda x} \\, dx$ does not converge as a [[Lebesgue integral|DWJF7ABPR1]].\nSuch integrals are still given a meaning as distributions, which is the setting of the more general oscillatory integrals of Hörmander.\n\nIf $x_0$ is the only critical point in the support of $\\psi$ and the Hessian of $\\phi$ at $x_0$ is invertible, the method of stationary phase gives $I(\\lambda) = O(\\lambda^{-n/2})$ as $\\lambda \\to \\infty$.\n",
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