Definition:Laplace Transform

Definition
Let $f \left({t}\right): S \to \C$ or $S \to \R$ be a function of a real variable $t$, where $\left [{t \,.\,.\, +\infty} \right) \subseteq S \subseteq \R, t \ge 0$.

The Laplace transform of $f$, denoted $\mathcal L \left\{{f}\right\}$ or $F$, is defined as:


 * $\displaystyle \mathcal L \left\{{f\left({t}\right)}\right\} = F \left({s}\right) = \int_0^{\to +\infty} e^{-st}f\left({t}\right) \ \mathrm d t$

wherever this improper integral exists.

Here $\mathcal L \left\{{f}\right\}$ is a complex function of the variable $s$.

Notation
Also denoted as:


 * $\mathcal L \left[{f \left({t}\right)}\right]$


 * $\mathscr L \left\{ {f \left({t}\right)}\right\}$


 * $\tilde f \left({s}\right)$

Comment
Although the definition of the Laplace transform has $s$ be a complex variable, sometimes the restriction of $\mathcal L \left\{{f}\right\}\left({s}\right)$ to wholly real $s$ is sufficient to solve a particular differential equation.

Therefore, elementary textbooks introducing the Laplace transform will often write something like the following:


 * A profound understanding of the workings of the Laplace transform requires considering it to be a so-called analytic function of a complex variable, but in most of this book we shall assume that the variable $s$ is real.


 * -- : $\S 3.1$

Also see

 * Definition:Inverse Laplace Transform