World Library  
Flag as Inappropriate
Email this Article

Inverse Laplace transform

Article Id: WHEBN0000245688
Reproduction Date:

Title: Inverse Laplace transform  
Author: World Heritage Encyclopedia
Language: English
Subject: Impulse response, Missing science topics/ExistingMathI, List of transforms, Integral transforms, Thomas John I'Anson Bromwich
Publisher: World Heritage Encyclopedia

Inverse Laplace transform

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:

\mathcal{L}\{f\}(s) = \mathcal{L}\{f(t)\}(s) = F(s),

where \mathcal{L} 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.[1][2]

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


  • Mellin's inverse formula 1
  • Software tools 2
  • See also 3
  • References 4
  • External links 5

Mellin's inverse formula

An integral formula for the inverse Laplace transform, called the Mellin's inverse formula, the Bromwich integral, or the FourierMellin integral, is given by the line integral:

f(t) = \mathcal{L}^{-1} \{F\}(t) = \mathcal{L}^{-1} \{F(s)\}(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,

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, or F(s) is a smooth function on −∞ < Re(s) < ∞ (i.e., no singularities), 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.

Software tools

  • InverseLaplaceTransform performs symbolic inverse transforms in Mathematica
  • Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain in Mathematica gives numerical solutions[3]
  • ilaplace performs symbolic inverse transforms in MATLAB
  • Numerical Inversion of Laplace Transforms in Matlab

See also


  1. ^ Cohen, A. M. (2007). "Inversion Formulae and Practical Results". Numerical Methods for Laplace Transform Inversion. Numerical Methods and Algorithms 5. p. 23.  
  2. ^  
  3. ^ Abate, J.; Valkó, P. P. (2004). "Multi-precision Laplace transform inversion". International Journal for Numerical Methods in Engineering 60 (5): 979.  
  • Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York:  
  • Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London:  
  • Boas, Mary (1983), Mathematical Methods in the physical sciences, (p. 662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the fourier transform)  

External links

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

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

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.