Editing Two-sided Laplace transform (section)

{{Short description|Mathematical operation}}
{{more footnotes needed|date=September 2015}}
In [[mathematics]], the '''two-sided Laplace transform''' or '''bilateral Laplace transform''' is an [[integral transform]] equivalent to [[probability]]'s [[moment-generating function]].  Two-sided Laplace transforms are closely related to the [[Fourier transform]], the [[Mellin transform]], the [[Z-transform]] and the ordinary or one-sided [[Laplace transform]]. If ''f''(''t'') is a real- or complex-valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral
:<math>\mathcal{B}\{f\}(s) = F(s) = \int_{-\infty}^\infty e^{-st} f(t)\, dt.</math>

The integral is most commonly understood as an [[improper integral]], which converges [[if and only if]] both integrals
:<math>\int_0^\infty e^{-st} f(t) \, dt,\quad \int_{-\infty}^0  e^{-st} f(t)\, dt</math>

exist.  There seems to be no generally accepted notation for the two-sided transform; the 
<math>\mathcal{B}</math> used here recalls "bilateral". The two-sided transform
used by some authors is
:<math>\mathcal{T}\{f\}(s) = s\mathcal{B}\{f\}(s) = sF(s) = s \int_{-\infty}^\infty  e^{-st} f(t)\, dt.</math>

In pure mathematics the argument ''t'' can be any variable, and Laplace transforms are used to study how [[differential operator]]s transform the function.

In [[science]] and [[engineering]] applications, the argument ''t'' often represents time (in seconds), and the function ''f''(''t'') often represents a [[signal (information theory)|signal]] or waveform that varies with time.  In these cases, the signals are transformed by [[Filter (signal processing)|filters]], that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time ''t'' cannot depend on an output which is a higher value of ''t''.
In population ecology, the argument ''t'' often represents spatial displacement in a dispersal kernel.

When working with functions of time, ''f''(''t'') is called the '''time domain''' representation of the signal, while ''F''(''s'') is called the '''s-domain''' (or ''Laplace domain'') representation. The inverse transformation then represents a ''synthesis'' of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the ''analysis'' of the signal into its frequency components.