Editing Two-sided Laplace transform (section)

==Relationship to the Fourier transform==
The [[Fourier transform]] can be defined in terms of the two-sided Laplace transform:

:<math>\mathcal{F}\{f(t)\} = F(s = i\omega) = F(\omega).</math>

Note that definitions of the Fourier transform differ, and in particular
:<math>\mathcal{F}\{f(t)\} = F(s = i\omega) = \frac{1}{\sqrt{2\pi}} \mathcal{B}\{f(t)\}(s)</math>
is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as
:<math>\mathcal{B}\{f(t)\}(s) = \mathcal{F}\{f(t)\}(-is).</math>

The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip <math>a < \Im(s) < b</math> which may not include the real axis where the Fourier transform is supposed to converge.

This is then why Laplace transforms retain their value in control theory and signal processing: the convergence of a Fourier transform integral within its domain only means that a linear, shift-invariant system described by it is stable or critical. The Laplace one on the other hand will somewhere converge for every impulse response which is at most exponentially growing, because it involves an extra term which can be taken as an exponential regulator. Since there are no superexponentially growing linear feedback networks, Laplace transform based analysis and solution of linear, shift-invariant systems, takes its most general form in the context of Laplace, not Fourier, transforms.

At the same time, nowadays Laplace transform theory falls within the ambit of more general [[integral transform]]s, or even general [[harmonic analysis]]. In that framework and nomenclature, Laplace transforms are simply another form of Fourier analysis, even if more general in hindsight.