Post's inversion formula
Encyclopedia
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.
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
Emil Leon Post
Emil Leon Post was a mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.-Early work:...
, 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.
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
-
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 home 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 differintegralGrunwald-Letnikov differintegralIn mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus, that allows one to take the derivative a non-integer number of times...
to evaluate the derivatives.
Post inversion has attracted 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 it interesting to calculate the asymptotic behaviour for big 'x' using inverse Mellin transformMellin transformIn mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
s for several arithmetical functions related to the Riemann HypothesisRiemann hypothesisIn mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
.