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  "body": "\n## Summary\n\nThe strip of convergence of a [[two-sided Laplace transform|6J2AGD0BYY]] is the vertical strip $\\alpha \u003c \\operatorname{Re} s \u003c \\beta$ in the complex plane where the defining integral converges absolutely.\nThe left edge $\\alpha$ is set by the growth of $f$ as $t \\to +\\infty$, and the right edge $\\beta$ by its growth as $t \\to -\\infty$.\nInside the strip the transform is holomorphic.\nThe strip must be given along with the formula, since one formula can be the transform of different functions on different strips.\n\n## Explanations\n\n### L1 human\n\nThe two-sided transform needs $f(t) e^{-s t}$ to be small at both ends of the time axis.\nA large $\\operatorname{Re} s$ helps for positive times and hurts for negative times, so only a band of values of $\\operatorname{Re} s$ works.\nFor $e^{-\\abs{t}}$ the band is $-1 \u003c \\operatorname{Re} s \u003c 1$.\nThe formula $1/(1 - s)$ is the transform of $e^{t}$ for $t \u003c 0$ (and $0$ for $t \u003e 0$) on $\\operatorname{Re} s \u003c 1$, but of $-e^{t}$ for $t \u003e 0$ (and $0$ for $t \u003c 0$) on $\\operatorname{Re} s \u003e 1$.\n",
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  "msc": [
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  "name": "Strip of convergence of the bilateral transform",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/HYA/HYA7QQ1E38.md",
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  "url": "https://tamnd.github.io/math2/wiki/strip-of-convergence-of-the-bilateral-transform/",
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