# 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.