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  "body": "\n## Definition\n\nLet $L_n$ be the [[Laguerre polynomial|WAMRHWW82R]] of degree $n$, and let $\\ell_n(t) = e^{-t/2} L_n(t)$ for $t \\ge 0$.\nThe functions $\\ell_0, \\ell_1, \\dots$ form an orthonormal basis of the [[space|SASP14MA1Z]] $L^2(0, \\infty)$.\nLet $f \\in L^2(0, \\infty)$, with coefficients $a_n = \\int_0^\\infty f(t)\\, \\ell_n(t)\\, dt$, and let $F$ be its [[Laplace transform|0V49JEZW06]].\nThe *Laguerre series expansion* of $F$ is\n$$F(s) = \\sum_{n=0}^{\\infty} a_n \\frac{(s - 1/2)^n}{(s + 1/2)^{n+1}}, \\qquad \\operatorname{Re} s \u003e 0.$$\n\n## Remarks\n\nThe expansion holds because the Laplace transform of $\\ell_n$ is $(s - 1/2)^n / (s + 1/2)^{n+1}$, and because $F(s)$ is the $L^2$ inner product of $f$ with $e^{-\\bar{s} t}$, which lies in $L^2(0, \\infty)$ when $\\operatorname{Re} s \u003e 0$.\nWith $w = (s - 1/2)/(s + 1/2)$, which maps the half-plane $\\operatorname{Re} s \u003e 0$ onto the unit disk, it reads $(s + 1/2) F(s) = \\sum_{n} a_n w^n$.\nThe coefficients satisfy $\\sum_n \\abs{a_n}^2 = \\norm{f}_2^2 \u003c \\infty$.\n\nExample: $f = \\ell_1$, that is $f(t) = e^{-t/2}(1 - t)$.\nThen $a_1 = 1$ and all other $a_n = 0$.\nThe check: $\\int_0^\\infty e^{-st} e^{-t/2}(1 - t)\\, dt = \\frac{1}{s + 1/2} - \\frac{1}{(s + 1/2)^2} = \\frac{s - 1/2}{(s + 1/2)^2}$.\n\nNon-example: $f(t) = 1$ has the Laplace transform $1/s$, but $f \\notin L^2(0, \\infty)$, so the coefficients $a_n$ are not defined by the formula above.\n\nApplying the same expansion to $e^{-\\sigma t} f(bt)$ for parameters $\\sigma$ and $b \u003e 0$ gives the scaled variants used in practice.\n",
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