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For one-sided Laplace transforms I can find many algorithms to invert them numerically (e.g. algorithms named after: Talbot, Stehfest, Euler, ...).

However, I am interested in numerical inversion of bilateral Laplace transforms:

$\hat{f}(s)=\int_{-\infty}^\infty f(t) e^{-s t} {\mathrm d}t.$

How can this be done?

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The usual way, I think. The inverse transformation is given by Fourier transforming any vertical slices of $\hat{f}$ within the region of convergence so that the usual numerical methods for Fourier transforms apply.

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    $\begingroup$ Thank you, Johannes. Let me check if I understand. If we would consider $\hat{f}(si)$ (for $si$ in the region of convergence of $\hat{f}$), then the bilateral LT is also a FT. Then we invert the FT. Could you point me to "the usual way"? $\endgroup$
    – David
    Commented Feb 21, 2018 at 16:42

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