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  "body": "\n## Summary\n\nAn oscillatory integral is an integral of the form $\\int e^{i\\lambda\\phi(x)} \\psi(x) \\, dx$.\nHere $\\phi$ is a real phase function, $\\psi$ is an amplitude and $\\lambda$ is a large real parameter.\nThe main question is how fast it decays as $\\lambda \\to \\infty$, which depends on the critical points of $\\phi$.\nFourier transforms of measures on curved surfaces, Bessel functions and solutions of dispersive equations are all oscillatory integrals.\nTheir estimates are basic tools in harmonic analysis, number theory and PDE.\n\n## Explanations\n\n### L1 human\n\nWhen $\\lambda$ is large, the factor $e^{i\\lambda\\phi(x)}$ spins around the unit circle very fast as $x$ moves.\nIts positive and negative parts nearly cancel, so the integral is small.\nThe cancellation is weakest where $\\phi$ is flat, at points where its derivative vanishes, and these points decide the size of the integral.\nFor example, $\\int_{-\\infty}^{\\infty} e^{i\\lambda x^2} e^{-x^2} \\, dx = \\sqrt{\\pi/(1 - i\\lambda)}$, which is of size $\\lambda^{-1/2}$ for large $\\lambda$, because the phase $x^2$ is flat at $x = 0$.\n",
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