Inverse Laplace transform

In mathematics, the inverse Laplace transform of F(s) is the function f(t) which has the property

where is the Laplace transform.

It can be proven, that if a function F(s) has the inverse Laplace transform f(t), i.e. f is a piecewise-continuous and exponentially-restricted real function f satisfying the condition

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).

The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamic systems.

Integral form

An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, 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). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, then γ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.

In practice, computing the complex integral can be done by using the Cauchy residue theorem.

It is named after Hjalmar Mellin, Joseph Fourier, and Thomas John I'Anson Bromwich.

Post's inversion formula

An alternative formula for the inverse Laplace transform is given by Post's inversion formula.

See also

• Inverse Fourier transform

see again in laplace transform

References

• Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95314-4 
• Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London: CRC Press, ISBN 978-0-8493-2876-3 

External links

• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.

• Rational inversion of the Laplace Transform

This article incorporates material from Mellin's inverse formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.