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:
- $\LL \sqbrk {\map f t}$
- $\mathscr L \set {\map f t}$
- $\mathbf L \map f t$
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