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:


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

whenever this improper integral exists.

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

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

Let $f$ be discontinuous or not defined at $t = 0$.

Then the Laplace transform of $f$ is defined as:


 * $\displaystyle \mathcal L \left\{{f \left({t}\right)}\right\} = F \left({s}\right) = \int_{0^+}^{\to +\infty} e^{-s t} f \left({t}\right) \ \mathrm d t = \lim_{\epsilon \to 0^+}\int_{\epsilon}^{\to +\infty} e^{-s t} f \left({t}\right) \ \mathrm d t$

whenever this improper integral exists.

Here the integral is improper both because of its lower bound and because of its upper bound.

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