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  "body": "\n## Summary\n\nThe [[Fourier transform|DPVM34ADZV]] of a function on $\\R$ is the [[two-sided Laplace transform|6J2AGD0BYY]] of the same function evaluated on the imaginary axis.\nWith $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i x \\xi} \\, dx$, this reads $\\hat f(\\xi) = \\mathcal{B}f(2\\pi i \\xi)$.\nMore generally, the values of $\\mathcal{B}f$ on a vertical line $\\operatorname{Re} s = \\sigma$ are the Fourier transform of $e^{-\\sigma x} f(x)$.\nThis lets results on one transform pass to the other, such as inversion formulas, Plancherel's theorem and the use of analytic continuation.\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform uses the weights $e^{-sx}$ with a complex number $s$.\nWhen $s$ is purely imaginary, these weights are the pure oscillations $e^{-i\\omega x}$ of the Fourier transform.\nSo the Fourier transform is the Laplace transform seen along one vertical line of the complex plane.\nFor example, $f(x) = e^{-\\abs{x}}$ has $\\mathcal{B}f(s) = 2/(1 - s^2)$ for $\\abs{\\operatorname{Re} s} \u003c 1$, and putting $s = 2\\pi i \\xi$ gives $\\hat f(\\xi) = 2/(1 + 4\\pi^2 \\xi^2)$.\n",
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