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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be [[integrable|DWJF7ABPR1]] on each bounded interval.\nThe *two-sided Laplace transform* of $f$ is the function\n$$\\mathcal{B}\\{f\\}(s) = \\int_{-\\infty}^{\\infty} f(t) \\, e^{-s t} \\, dt = \\lim_{T_1, T_2 \\to \\infty} \\int_{-T_1}^{T_2} f(t) \\, e^{-s t} \\, dt.$$\nHere $T_1$ and $T_2$ tend to $\\infty$ independently.\nThe transform is defined at each complex $s$ for which this [[improper integral|05KQ3F99BQ]] converges.\n\n## Remarks\n\nExample: for $f(t) = e^{-\\abs{t}}$, the part over $[0, \\infty)$ is $1/(1 + s)$ for $\\operatorname{Re} s \u003e -1$, and the part over $(-\\infty, 0]$ is $1/(1 - s)$ for $\\operatorname{Re} s \u003c 1$.\nSo $\\mathcal{B}\\{f\\}(s) = \\frac{2}{1 - s^2}$ for $-1 \u003c \\operatorname{Re} s \u003c 1$.\nNon-example: for the constant $f = 1$, the part over $[0, \\infty)$ needs $\\operatorname{Re} s \u003e 0$ and the part over $(-\\infty, 0]$ needs $\\operatorname{Re} s \u003c 0$, so the integral converges for no $s$.\nIf $f(t) = 0$ for $t \u003c 0$, the two-sided transform is the one-sided [[Laplace transform|05XT5AVJBK]] of $f$ restricted to $[0, \\infty)$.\nFor a random variable $X$ with density $p$, $\\mathcal{B}\\{p\\}(-s) = \\mathbb{E}[e^{s X}]$ is its moment generating function wherever it converges.\n",
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