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  "body": "\n## Example\n\nLet $\\alpha \u003e -1$ be real and $f(t) = t^\\alpha$ for $t \u003e 0$.\nIts [[Laplace transform|05XT5AVJBK]] is\n$$\\mathcal{L}\\{t^\\alpha\\}(s) = \\frac{\\Gamma(\\alpha + 1)}{s^{\\alpha + 1}} \\quad (\\operatorname{Re} s \u003e 0),$$\nwhere $\\Gamma$ is the [[gamma function|0V3FZMVB4K]] and $s^{\\alpha + 1}$ is the principal power.\nFor $\\alpha = n \\in \\N$ this is $\\mathcal{L}\\{t^n\\}(s) = n!/s^{n + 1}$, with $n!$ the [[factorial|0WFCGVMRRC]].\nCheck for real $s \u003e 0$: the substitution $u = st$ gives\n$$\\int_0^\\infty t^\\alpha e^{-st}\\, dt = \\frac{1}{s^{\\alpha + 1}} \\int_0^\\infty u^\\alpha e^{-u}\\, du = \\frac{\\Gamma(\\alpha + 1)}{s^{\\alpha + 1}}.$$\nThe integral converges at $0$ because $\\alpha \u003e -1$, and at $\\infty$ because of the factor $e^{-st}$.\nBoth sides are holomorphic in $s$ on $\\operatorname{Re} s \u003e 0$ and agree for real $s \u003e 0$, so they agree on the whole half-plane.\nFor example, $\\mathcal{L}\\{t^{-1/2}\\}(s) = \\sqrt{\\pi / s}$, since $\\Gamma(1/2) = \\sqrt{\\pi}$.\n\n## Explanations\n\n### L1 human\n\nPowers of $t$ become negative powers of $s$: $1 \\mapsto 1/s$, $t \\mapsto 1/s^2$, $t^2 \\mapsto 2/s^3$, and so on.\nThe factorials come from integrating by parts once for each power of $t$.\nFor powers that are not whole numbers the factorial is replaced by the gamma function.\n",
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