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.

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