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Concept
Inverse Laplace transform
Summary
In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property:
where denotes the Laplace transform.
It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.
The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems.
An integral formula for the inverse Laplace transform, called the Mellin's inverse formula, the Bromwich integral, or the Fourier–Mellin integral, is given by the line integral:
where the integration is done along the vertical line Re(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F(s) and F(s) is bounded on the line, for example if the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is an entire function , then γ can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.
In practice, computing the complex integral can be done by using the Cauchy residue theorem.
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
for t > 0, where F(k) is the k-th derivative of F with respect to s.
Official source
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