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. Any algorithm for the "Inverse Laplace Transform" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "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.) or any database download or API access for "Inverse Laplace Transform" or any other element are not public (except explicit open source licence like Creative Commons). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Inverse Laplace Transform" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source.
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Inverse Laplace Transform on dCode.fr [online website], retrieved on 2025-04-16,
