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  "body": "\n## Statement\n\nLet $\\mu \\in \\mathbb{R}$, let $\\sigma^2 \u003e 0$ and let $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$. Then for each $t \\in \\mathbb{R}$,\n$$\\varphi_X(t) = E\\left[e^{itX}\\right] = \\exp\\left(i \\mu t - \\tfrac{1}{2} \\sigma^2 t^2\\right).$$\nIn particular, the standard normal distribution has $\\varphi(t) = e^{-t^2/2}$.\n\n## Remarks\n\nOne proof: the [[moment-generating function|0B1G4E21G7]] $E[e^{zX}] = \\exp(\\mu z + \\sigma^2 z^2/2)$ holds for real $z$. Both sides are entire functions of $z$, so the identity extends to all complex $z$, and one puts $z = it$.\nAnother proof: $\\varphi(t) = e^{-t^2/2}$ for $\\mathcal{N}(0, 1)$ is the solution of $\\varphi'(t) = -t \\varphi(t)$ with $\\varphi(0) = 1$. Integration by parts gives this differential equation.\nSo the standard normal density is, up to the factor $\\sqrt{2\\pi}$, its own Fourier transform.\n\n## Explanations\n\n### L1 human\n\nThe characteristic function of a normal variable is again a bell-shaped function.\nA narrow normal distribution, with small $\\sigma$, has a wide characteristic function, and conversely.\n",
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