# Laplace Transform Symbol in LaTeX

## 1. What is Laplace transform?

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 $$F(s)$$.

## 2. Laplace symbol in LaTeX

From above you may remark that we used the symbol $$\mathscr{L}\{f(t)\}$$ 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:

% Laplace transform symbol in LaTeX
\documentclass{article}

% Required package
\usepackage{mathrsfs}

\begin{document}

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

\end{document}

## 3. Another version of Laplace symbol

Some documents prefer to use the symbol $$\mathcal{L}\{f(t)\}$$ 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:

% Second version of Laplace transform symbol in LaTeX
\documentclass{article}

% Required package
\usepackage{amsmath}

\begin{document}

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

\end{document}

Compiling this code yields:

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

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