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Inverse Laplace Transform

Tool to calculate the inverse Laplace transform of a function, transformation widely used for the analysis of linear dynamical systems.

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Inverse Laplace Transform -

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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}

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Inverse Laplace Transform on dCode.fr [online website], retrieved on 2024-12-19, https://www.dcode.fr/inverse-laplace-transform

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