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  "body": "\n## Definition\n\nLet $f : [0, \\infty) \\to \\mathbb{C}$ be a [[function|0EPAT51DXQ]] that is integrable on each interval $[0, T]$ with $T \u003e 0$.\nThe Laplace transform of $f$ is the function $F = \\mathcal{L}\\{f\\}$ given by\n$$F(s) = \\int_0^\\infty f(t)\\, e^{-s t}\\, dt = \\lim_{T \\to \\infty} \\int_0^T f(t)\\, e^{-s t}\\, dt,$$\nfor each complex number $s$ for which the limit exists.\n\n## Remarks\n\nLet $f$ be of exponential order: there are constants $M, a$ with $|f(t)| \\le M e^{a t}$ for all $t \\ge 0$. Then the integral converges absolutely for $\\operatorname{Re} s \u003e a$.\nSome transforms: $\\mathcal{L}\\{1\\}(s) = 1/s$ and $\\mathcal{L}\\{e^{a t}\\}(s) = 1/(s - a)$ for $\\operatorname{Re} s \u003e \\operatorname{Re} a$, and $\\mathcal{L}\\{\\sin \\omega t\\}(s) = \\omega / (s^2 + \\omega^2)$ for $\\operatorname{Re} s \u003e 0$ and real $\\omega$.\nThe transform is linear: $\\mathcal{L}\\{c f + g\\} = c \\mathcal{L}\\{f\\} + \\mathcal{L}\\{g\\}$ where both sides converge.\nIt turns derivatives into multiplication by $s$. See the [[Laplace transform of a derivative|0K090GVHVN]].\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform turns a function of time into a function of a complex variable $s$.\nIt weights the function by $e^{-st}$, which decays fast, and adds up the result over all positive times.\nIts main use is to turn linear differential equations into algebraic equations.\n",
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