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 transforms, or even general harmonical analysis. In that framework and nomenclature, Laplace transforms are simply another form of Fourier analysis, even if more general in hindsight. Relationship to other integral transforms If u is the Heaviside step function, equal to zero when its argument is less than zero, to one-half when its argument equals zero, and to one when its argument is greater than zero, then the Laplace transform may be defined in terms of the two-sided Laplace transform by On the other hand, we also have where is the function that multiplies by minus one (), so either version of the Laplace transform can be defined in terms of the other. The Mellin transform may be defined in terms of the two-sided Laplace transform by with as above, and conversely we can get the two-sided transform from the Mellin transform by The moment-generating function of a continuous probability density function ƒ(x) can be expressed as . Properties The following properties can be found in and Most properties of the bilateral Laplace transform are very similar to properties of the unilateral Laplace transform, but there are some important differences: Parseval's theorem and Plancherel's theorem Let and be functions with bilateral Laplace transforms and in the strips of convergence . Let with . Then Parseval's theorem holds: This theorem is proved by applying the inverse Laplace transform on the convolution theorem in form of the cross-correlation. Let be a function with bilateral Laplace transform in the strip of convergence . Let with . Then the Plancherel theorem holds: Uniqueness For any two functions for which the two-sided Laplace transforms exist, if i.e. for every value of then almost everywhere. Region of convergence Bilateral transform requirements for convergence are more difficult than for unilateral transforms. The region of convergence will be normally smaller. If f is a locally integrable function (or