Post's inversion formula

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 with respect to s.

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 personal computers, the main efforts to use this formula have come from dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the Grunwald-Letnikov differintegral to evaluate the derivatives.

Post's inversion has attracted interest due to the improvement in computational science and the fact that it is not necessary to know where the poles of F(s) lie, which make it possible to calculate the asymptotic behaviour for big x using inverse Mellin transforms for several arithmetical functions related to the Riemann Hypothesis.

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