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 $\hat{C}(\beta )$. The necessary input is only the values of $\hat{C}(\beta )$ on the positive real axis. The method is applicable provided that the functions $\hat{C}(\beta )$ belong to the function space $L_{\alpha }^2$ defined by the condition that $G(x)=e^{x\alpha }\hat{C}(e^x),\alpha >0$ has to be square integrable. This space includes sums of exponential decays $\hat{C}(\beta )={\sum }_n^{\infty }{a}_n{e}^{-\beta E_n}$, 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.

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