# Comparison of the Laplace and Fourier Transform
Both the [[Laplace transform]] and the [[Fourier transform]] transform a function of a real variable, usually $t$ in the *time domain*, into a complex function.
The Fourier transform generates a complex function of a *real* variable, $\omega$, and the Laplace transform generates a complex function of a *complex* variable, $s$.
$s=\sigma+i\omega$
## Definitions
The typical notion of the Laplace transform is the *unilateral* transform, which is *only* defined for $t\ge 0$.
The Fourier transform is defined for *all* $t$.
## Relationship
The Fourier transform is a *special case* of the unilateral Laplace transform, defined *only* on the $i\omega$ axis of the $s$ plane.
For a function $f(t)$ that is *only* non-zero for $t\ge 0$,
$\mathcal{F}[f(t)]=\mathcal{L}[f(t)]\Bigr|_{s=i\omega}$
## Applications
The Laplace transform is applicable to *more functions* than the Fourier transform.
A significant advantage is the Laplace transform of *integral* and *differential* equations into *polynomials*, allowing for the analysis of *dynamic systems* even with *initial condition* constraints.
Fourier transforms are only applicable for *steady systems* but provide better insight into the frequency characteristics of the system.