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  "body": "\n## Summary\n\nThe Laguerre series expansion writes the [[Laplace transform|0V49JEZW06]] $F(s)$ of a function, times $s + 1/2$, as a power series in $w = (s - 1/2)/(s + 1/2)$.\nHere $w$ is a Möbius transformation of the transform variable $s$.\nIts coefficients are the coefficients of the function in the orthonormal basis of Laguerre functions $e^{-t/2} L_n(t)$ of $L^2(0, \\infty)$.\nHere $L_n$ are the [[Laguerre polynomials|WAMRHWW82R]].\nThe expansion converts Laplace inversion into the computation of Taylor coefficients of an analytic function on the unit disk.\nIt is the basis of Weeks' method for the numerical inversion of the Laplace transform.\n\n## Explanations\n\n### L1 human\n\nThe Laplace transform of each Laguerre function is very simple: a power of $(s - 1/2)$ divided by a power of $(s + 1/2)$.\nSo if a function is a sum of Laguerre functions, its Laplace transform is a sum of these simple terms.\nThe change of variable $w = (s - 1/2)/(s + 1/2)$ makes this sum an ordinary power series in $w$.\nFor example, $f(t) = e^{-t/2}$ is the first Laguerre function, and its transform $1/(s + 1/2)$ is the first term of the series.\nTo recover $f$ from its transform, one reads off the power series coefficients in $w$.\n",
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