Tool to calculate the inverse Laplace transform of a function, transformation widely used for the analysis of linear dynamical systems.
Inverse Laplace Transform - dCode
Tag(s) : Functions
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Inverse Laplace Transform
Laplace Inverse Transform Calculator
Answers to Questions (FAQ)
What is the Laplace Inverse Transform? (Definition)
The Laplace inverse transformation of a function $ F $ is denoted $ \mathcal{L}^{-1} $ (or sometimes $ F^{-1} $), its result is called the inverse Laplace transform (ILT).
For any function $ F(s) $ with $ s \in \mathbb{C} $, the Laplace transform of real variable $ t \in \mathbb{R} $ is:
$$ \mathcal{L}^{-1}(t) = \frac{1}{2 i \pi} \int_{\gamma - i \cdot \infty}^{\gamma + i \cdot \infty} \exp(st) F(s) \, {\rm d} s $$
with $ \gamma $ a constant chosen such as the integration avoids all singularities of $ F(s) $.
Sometimes the transform is denoted $ \mathcal{L}^{-1}[F(s)](t) $.
In Europe, the complex variable $ s $ is sometimes noted $ p $.
If $ \gamma = 0 $ then the inverse Laplace transform is identical to the inverse Fourier transform.
How to calculate the inverse Laplace transform?
The calculation of the inverse Laplace transform is an integral calculation (see definition above).
On dCode, indicate the function, its complex variable (often $ s $ or $ p $), and the real variable (often $ t $ or $ x $).
Example: $ F(s) = 1/(1-s) $ and $ \mathcal{L}^{-1}[F(s)](t) = -\exp(t) $.
How to write Inverse Laplace transform?
The Laplace transform is written with a handwritten L, to the power -1: $ \mathcal{L}^{-1} $
In LaTeX, use \mathcal{L}^{-1}
Source code
dCode retains ownership of the "Inverse Laplace Transform" source code. Except explicit open source licence (indicated Creative Commons / free), the "Inverse Laplace Transform" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Inverse Laplace Transform" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Inverse Laplace Transform" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
The copy-paste of the page "Inverse Laplace Transform" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Inverse Laplace Transform on dCode.fr [online website], retrieved on 2024-11-18,
