A new method for numerical inversion of the Laplace transform

Abstract

A formula of Doetsch ({\em Math. Zeitschr.} {\bf 42}, 263 (1937)) is generalized and used to numerically invert the one-sided Laplace transform ${\hat C}(β)$. The necessary input is only the values of ${\hat C}(β)$ on the positive real axis. The method is applicable provided that the functions $\hat{C}(β)$ belong to the function space $L^2_α$ defined by the condition that $ G(x) = e^{xα}\hat{C}(e^x),~ α> 0$ has to be square integrable. This space includes sums of exponential decays ${\hat C}(β)=\sum_n^{\infty}a_n e^{-βE_n}$, e.g. partition functions with $a_n = 1$. In practice, the inversion algorithm consists of two subsequent fast Fourier transforms. High accuracy inverted data can be obtained, provided that the signal is also highly accurate. The method is demonstrated for a harmonic partition function and resonant transmission through a barrier. We find accurately inverted functions even in the presence of noise.