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  "body": "\n## Statement\n\nLet $\\mu$ be a [[probability measure|059TJN0NZ1]] on $\\R$.\nLet $I$ be the set of real $\\sigma$ with $\\int_{\\R} e^{-\\sigma x} \\, d\\mu(x) \u003c \\infty$.\nThen:\n\n1. $I$ is an interval that contains $0$.\n2. For every $s \\in \\C$ with $\\operatorname{Re} s \\in I$, the integral $\\mathcal{B}\\mu(s) = \\int_{\\R} e^{-sx} \\, d\\mu(x)$ converges absolutely.\n3. On the open strip $S = \\set{s \\in \\C : \\operatorname{Re} s \\in \\operatorname{int} I}$, the function $\\mathcal{B}\\mu$ is [[holomorphic|0NAHXR1FRH]], and for every $n \\ge 0$ and $s \\in S$,\n$$(\\mathcal{B}\\mu)^{(n)}(s) = \\int_{\\R} (-x)^n e^{-sx} \\, d\\mu(x),$$\nwhere this integral converges absolutely.\n\nHere $\\operatorname{int} I$ is the interior of $I$, and $\\mathcal{B}\\mu$ is the [[two-sided Laplace transform|6J2AGD0BYY]] of $\\mu$.\n\n## Remarks\n\nPart 1 holds because $\\sigma \\mapsto e^{-\\sigma x}$ is convex, so $e^{-\\sigma x} \\le e^{-a x} + e^{-b x}$ for $a \\le \\sigma \\le b$; and $0 \\in I$ because $\\mu$ has total mass $1$.\nPart 2 holds because $\\abs{e^{-sx}} = e^{-x \\operatorname{Re} s}$.\nFor part 3, fix $a \u003c \\operatorname{Re} s \u003c b$ in $I$; then $\\abs{x}^n e^{-x \\operatorname{Re} s} \\le C_n (e^{-a x} + e^{-b x})$ for a constant $C_n$, so [[differentiation under the integral sign|0AR06W1E1B]] is allowed by dominated convergence.\nOn the real axis this says that the [[moment-generating function|0WQVXP7PDB]] $M(t) = \\mathcal{B}\\mu(-t)$ is real analytic on the interior of its domain, with $M^{(n)}(t) = \\int x^n e^{tx} \\, d\\mu(x)$.\nExample: for the exponential law with rate $\\lambda$, $I = (-\\lambda, \\infty)$ and $\\mathcal{B}\\mu(s) = \\lambda/(\\lambda + s)$, which is holomorphic for $\\operatorname{Re} s \u003e -\\lambda$ and has a pole at $s = -\\lambda$.\nThe open strip is needed in part 3: the derivative formula uses room on both sides of $\\operatorname{Re} s$, and at an end point of $I$ the integral $\\int \\abs{x} e^{-x \\operatorname{Re} s} \\, d\\mu(x)$ may diverge.\nIf $I = \\set{0}$, as for the Cauchy law, the strip is empty and the statement says nothing.\n\n## Explanations\n\n### L1 human\n\nAveraging $e^{-sX}$ over a random quantity $X$ gives a function of $s$.\nThis theorem says that, wherever the average is finite with a little room to spare, the function is as smooth as possible.\nIt is given locally by a convergent power series.\nIts derivatives are found by differentiating inside the average.\n\n### L2 human\n\nThis is the reason why a moment-generating function that is finite near $0$ produces all the moments, as its derivatives at $0$.\nIt is also the starting point of large deviation theory, where the convex function $\\log M$ and its derivatives are used.\nThe domain of finiteness is the [[strip of convergence|HYA7QQ1E38]] of the bilateral transform.\nThe same proof gives analyticity of the two-sided Laplace transform of any measure on its open strip.\nOn the imaginary axis the transform is the characteristic function, which is always defined but need not be analytic.\n",
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