The Laplace transform of an expression \(f(t)\) is denoted by \(\mathscr{L}\{f(t)\}\) and is defined as the semi infinite integral:

\(\displaystyle\mathscr{L}\{f(t)\}=\int_{t=0}^{\infty}f(t)e^{-st}dt\)

The variable \(s\) used on the right-hand side is a parameter, which is assumed as positive and large enough to ensure that the integral is convergent.

After solving the integral on the right-hand side of the above equation, the resulting value will be in the form of \(s\), which is known as the Laplace transform of \(f(t)\) and denoted \(F(s)\).

## Typesetting the Laplace symbol in LaTeX

From above you may remark that we used the symbol by \(\mathscr{L}\{f(t)\}\) to denote the Laplace transform and you may ask which command in LaTeX produces it? The Laplace transform symbol in LaTeX can be obtained using the command `\mathscr{L}`

provided by `mathrsfs`

package.

The above semi-infinite integral is produced in LaTeX as follows:

```
\usepackage{mathrsfs}
\begin{document}
\[
\mathscr{L}\{f(t)\}=\int_{t=0}^{\infty}f(t)e^{-st}dt
\]
\end{document}
```

provides the output like:

\(\displaystyle\mathscr{L}\{f(t)\}=\int_{t=0}^{\infty}f(t)e^{-st}dt\)

## Another version of the Laplace symbol

Some documents prefer to use the symbol \(\mathscr{L}\{f(t)\}\) to denote the Laplace transform of the function \(f(t)\). In LaTeX, this Laplace transform symbol can be obtained using the command `\mathcal{L}`

provided by `amsmath`

package. The above code becomes:

```
\usepackage{amsmath}
\begin{document}
\[
\mathcal{L}\{f(t)\}=\int_{t=0}^{\infty}f(t)e^{-st}dt
\]
\end{document}
```

produces an output like:

\(\mathcal{L}\{f(t)\}=\int_{t=0}^{\infty}f(t)e^{-st}dt\)

The Laplace transform symbol is an important mathematical symbol used to represent the Laplace transform of a function, and can be easily written in LaTeX using the `\mathcal{L}`

command or other available options.