Definition:Laplace Transform

Definition
Let $f: \left [{0 \,.\,.\, \to} \right) \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \left\{ {\R, \C}\right\}$.

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


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

wherever this improper integral exists.

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

Application in Physics
In the field of Signal Processing, the domain of $f$ is called the time domain, and the domain of $\mathcal L \left\{{f}\right\}$ is called the frequency domain.

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