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
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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.
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) $.
The Laplace transform is written with a handwritten L, to the power -1: $ \mathcal{L}^{-1} $
In LaTeX, use \mathcal{L}^{-1}
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Inverse Laplace Transform on dCode.fr [online website], retrieved on 2024-12-19,