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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be [[integrable|DWJF7ABPR1]] on each bounded interval.\nLet\n$$\\alpha = \\inf \\left\\{ \\sigma \\in \\R : \\int_0^\\infty \\abs{f(t)} e^{-\\sigma t} \\, dt \u003c \\infty \\right\\}, \\qquad \\beta = \\sup \\left\\{ \\sigma \\in \\R : \\int_{-\\infty}^0 \\abs{f(t)} e^{-\\sigma t} \\, dt \u003c \\infty \\right\\},$$\nwith $\\inf \\emptyset = +\\infty$ and $\\sup \\emptyset = -\\infty$.\nIf $\\alpha \u003c \\beta$, the *strip of convergence* of the [[two-sided Laplace transform|V94VYJ2RR1]] of $f$ is the open vertical strip\n$$S = \\{ s \\in \\C : \\alpha \u003c \\operatorname{Re} s \u003c \\beta \\}.$$\nIf $\\alpha \\ge \\beta$, the strip of convergence is empty.\n\n## Remarks\n\nThe integral defining $\\mathcal{B}\\{f\\}(s)$ converges absolutely for every $s \\in S$, and $\\mathcal{B}\\{f\\}$ is [[holomorphic|0DG0Z72CQD]] on $S$.\nThe integral may also converge on one or both boundary lines; these points are not counted in the open strip.\nExample: for $f(t) = e^{-\\abs{t}}$, one finds $\\alpha = -1$ and $\\beta = 1$, so $S = \\{-1 \u003c \\operatorname{Re} s \u003c 1\\}$.\nNon-example: for $f = 1$, $\\alpha = 0$ and $\\beta = 0$, so the strip is empty, and indeed the two-sided integral converges nowhere.\nThe strip is part of the data of a transform.\nThe function $1/(1 - s)$ on $\\operatorname{Re} s \u003c 1$ inverts to $e^t$ on $t \u003c 0$ and $0$ on $t \u003e 0$, while on $\\operatorname{Re} s \u003e 1$ it inverts to $-e^t$ on $t \u003e 0$ and $0$ on $t \u003c 0$.\n",
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  "msc": [
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  "name": "Strip of convergence of the bilateral transform",
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