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  "body": "\n## Definition\n\nLet $X$ be a real random variable with law $\\mu_X$. Let $i$ be the imaginary unit in the [[complex numbers|09FTAET5Z4]].\nThe characteristic function of $X$ is the function $\\varphi_X \\colon \\mathbb{R} \\to \\mathbb{C}$ with\n$$\\varphi_X(t) = E\\left[e^{itX}\\right] = E[\\cos(tX)] + i \\, E[\\sin(tX)] = \\int_{\\mathbb{R}} e^{itx} \\, \\mu_X(dx).$$\n\n## Remarks\n\nThe expectation exists for all $t$, because $|e^{itX}| = 1$. The second form uses [[Euler's formula|04R3CY41VJ]].\nAlways $\\varphi_X(0) = 1$, $|\\varphi_X(t)| \\le 1$, and $\\varphi_X$ is uniformly continuous on $\\mathbb{R}$.\nFor $a, b \\in \\mathbb{R}$, $\\varphi_{aX + b}(t) = e^{itb} \\varphi_X(at)$.\nIf $E[|X|^k] \u003c \\infty$, then $\\varphi_X$ is $k$ times continuously differentiable and $\\varphi_X^{(k)}(0) = i^k E[X^k]$.\n\n## Explanations\n\n### L1 human\n\nThe characteristic function is like the moment-generating function, but with an imaginary argument.\nThis small change makes it finite for every distribution, so it is a tool that always works.\n",
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