# Definition:Laplace Transform/Notation

## Notation for Laplace Transform

The function which serves as the argument of a Laplace transform is usually denoted by means of a lowercase letter, for example $f$, $g$, $y$, and so on.

The Laplace transform of this function is then denoted by the corresponding uppercase letter, that is $F$, $G$, $Y$, and so on.

Hence we have:

- $\laptrans {\map f t} = \map F s$

However, note that some sources reverse the cases of the symbols used to denote the functions under discussion:

- $\laptrans {\map F t} = \map f s$

Notation for the **Laplace transform** varies throughout the literature.

The notation preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ is:

- $\laptrans {\map f t} = \map F s$

Other notation that can be seen includes:

- $\mathcal L \sqbrk {\map f t}$

- $\mathscr L \set {\map f t}$

- $\mathbf L \map f t$

It is sometimes worth stressing the point that $\laptrans {\map f t}$ is a function of $s$ by expressing it as:

- $\map {\laptrans {\map f t} } s$

and this notation is occasionally seen on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources use a tilde $\tilde f$ to denote the Laplace transform.

Thus the Laplace transform of $\map u t$ is denoted $\map {\tilde u} t$.

However, this usage is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$ because the tilde does not present well in the version of the $\LaTeX$ renderer used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Sources

- 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Notation