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.
