Laguerre series expansion of the Laplace transform

Summary

The Laguerre series expansion writes the Laplace transform $F(s)$ of a function, times $s + 1/2$, as a power series in $w = (s - 1/2)/(s + 1/2)$. Here $w$ is a Möbius transformation of the transform variable $s$. Its coefficients are the coefficients of the function in the orthonormal basis of Laguerre functions $e^{-t/2} L_n(t)$ of $L^2(0, \infty)$. Here $L_n$ are the Laguerre polynomials. The expansion converts Laplace inversion into the computation of Taylor coefficients of an analytic function on the unit disk. It is the basis of Weeks' method for the numerical inversion of the Laplace transform.