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Laplace Transform

Tool to calculate the Laplace transform of an integrable function on R, the Laplace transform is denoted F or L.

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# Laplace Transform

## Laplace Transform Calculator

### What is the Laplace Tranform? (Definition)

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 $p$.

### How to calculate the Laplace transform?

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 $p$).

Example: $f(x) = \delta(t)$ and $\mathcal{L}(s) = 1$ with Dirac's $\delta$ function.

Example: $f(x) = U(t)$ and $\mathcal{L}(s) = \frac{1}{s^2}$ with the unit step function $U$.

On dCode the function $\delta()$ is noted dirac() or diracdelta()
The function $U()$ (unit step or Heaviside Theta function) is denoted unitstep() or heaviside() .
dCode also supports the rectangle() or gate() functions or the triangle() or hat() function.

### Why calculating the Laplace transform?

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.

### When applying the Laplace transform?

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.

### How to write Laplace transform?

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

In LaTeX, use \mathcal{L}

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