Post's inversion formula

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Post's inversion formula for Laplace transforms, named after Emil Post, is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.

The statement of the formula is as follows: Let f(t) be a continuous function on the interval [0, ∞) of exponential order, i.e.

\sup_{t>0} \frac{f(t)}{e^{bt}} < \infty

for some real number b. Then for all s > b, the Laplace transform for f(t) exists and is infinitely differentiable with respect to s. Furthermore, if F(s) is the Laplace transform of f(t), then the inverse Laplace transform of F(s) is given by

f(t) = \mathcal{L}^{-1} \{F(s)\} = \lim_{k \to \infty} \frac{(-1)^k}{k!} \left( \frac{k}{t} \right) ^{k+1} F^{(k)} \left( \frac{k}{t} \right)

for t > 0, where F(k) is the k-th derivative of F.

As can be seen from the formula, the need to evaluate derivatives of arbitrarily high orders renders this formula impractical for most purposes.


With the advent of powerful home computers, the main efforts using this formula , have come dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the Grunwald Letnikov differintegral to evaluate the derivatives.

Post inversion has attracted the interest due to the improvement in Computational Science and the fact that you don't need to know where the poles of F(s) lie , which make interesting to calculate the asymptotic behaviour for big 'x' using Mellin-inverse transforms for several Arithmetical functions dealing with Riemann Hypothesis


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