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  "body": "\n## Statement\n\nLet $f \\colon [0, \\infty) \\to \\C$ be integrable on each interval $[0, T]$, and let $F$ be its [[Laplace transform|05XT5AVJBK]].\nLet $a$ be a complex number, and let $s$ be a complex number such that the Laplace integral of $f$ converges at $s - a$.\nThen the Laplace integral of $t \\mapsto e^{a t} f(t)$ converges at $s$, and\n$$\\mathcal{L}\\{e^{a t} f(t)\\}(s) = F(s - a).$$\n\n## Remarks\n\nThe proof is one line: $e^{a t} f(t) e^{-s t} = f(t) e^{-(s - a) t}$, so the two integrals are the same integral.\nIn particular, the half-plane of convergence moves to the right by $\\operatorname{Re} a$.\nExample: from $\\mathcal{L}\\{\\sin \\omega t\\}(s) = \\omega / (s^2 + \\omega^2)$ one gets $\\mathcal{L}\\{e^{a t} \\sin \\omega t\\}(s) = \\omega / ((s - a)^2 + \\omega^2)$.\nLikewise $\\mathcal{L}\\{t^n e^{a t}\\}(s) = n! / (s - a)^{n + 1}$ for $\\operatorname{Re} s \u003e \\operatorname{Re} a$.\n\n## Explanations\n\n### L1 human\n\nMultiplying a signal by the [[exponential|0J0FCWV32D]] $e^{a t}$ just slides its transform sideways by $a$.\nSo a damped oscillation has the same transform as a plain oscillation, moved by the damping rate.\n\n### L2 human\n\nThis rule is also called the first shifting theorem or the $s$-shift rule.\nIt is the main tool for inverting terms such as $1/((s + 2)^2 + 9)$: complete the square, then read off $\\tfrac{1}{3} e^{-2t} \\sin 3t$.\nIt is the mirror image of the [[time shift theorem|M4VJHPZJMQ]], where a shift in $t$ becomes a factor $e^{-a s}$.\n",
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    "name": "Frequency shift theorem",
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  "msc": [
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  "name": "Frequency shift theorem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/CD3/CD35MSBNE7.md",
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