Laplace Transform

The Laplace transform is a mathematical technique that transforms a continuous time function into a complex variable function. This transformation simplifies the analysis of linear systems and their calculations.

The Laplace transformation of a function $ f $ is denoted $ \mathcal{L} $ (or sometimes $ F $), its result is called the Laplace transform.

For any function $ f(t) $ with $ t \in \mathbb{R} $, the Laplace transform of complex variable $ s \in \mathbb{C} $ is:

$$ \mathcal{L}(s) = \int_{0^-}^{+\infty}\exp(-pt) f(t) \, dt $$

Sometimes the transform is denoted $ \mathcal{L}[f(t)](s) $.

In Europe, the complex variable $ s $ can also be noted $ p $.

The calculation of the Laplace transform is an integral calculation (see definition above). The Laplace transform assumes that the initial values of functions are zero.

On dCode, indicate the function, its variable (often $ t $ or $ x $), and the complex variable (often $ s $ or $ p $).

The property of the Laplace transformation is to convert the integrals into division and the derivatives into multiplication.

$$ \mathcal{L} \{ f'(t) \} = s \int_{-\infty}^\infty e^{-st} f(t) \, dt = s \cdot \mathcal{L} \{ f(t) \} $$

This property makes it possible to solve linear differential equations.

The Laplace transformation has applications in many areas, including:

— Signal processing: For the filtering and analysis of signals, particularly in the field of communication.

— Electrical engineering: To analyze RLC electrical circuits and their transient response.

— Mechanics: To study the vibration response of mechanical systems and their damping.

— Process control: To design controllers and analyze the stability of industrial processes.

— etc.

The Laplace transform is written with a handwritten L $ \mathcal{L} $ (or sometimes with an uppercase f: $ F $)

In LaTeX, use \mathcal{L}