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| In [[mathematics]], the '''two-sided Laplace transform''' or '''bilateral Laplace transform''' is an [[integral transform]] closely related to the [[Fourier transform]], the [[Mellin transform]], and the ordinary or one-sided [[Laplace transform]]. If ''ƒ''(''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
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| :<math>\mathcal{B} \left\{f(t)\right\} = F(s) =
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| \int_{-\infty}^\infty e^{-st} f(t) \,dt.</math>
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| The integral is most commonly understood as an [[improper integral]], which converges if and only if each of the integrals
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| :<math>\int_0^\infty e^{-st} f(t) \, dt,\quad \int_{-\infty}^0 e^{-st} f(t) \, dt</math>
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| exists. There seems to be no generally accepted notation for the two-sided transform; the
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| <math>\mathcal{B}</math> used here recalls "bilateral". The two-sided transform
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| used by some authors is
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| :<math>\mathcal{T}\left\{f(t)\right\} = s\mathcal{B}\left\{f\right\} = sF(s) =
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| s \int_{-\infty}^\infty e^{-st} f(t) \, dt.</math>
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| In pure mathematics the argument ''t'' can be any variable, and Laplace transforms are used to study how [[differential operator]]s transform the function.
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| In [[science]] and [[engineering]] applications, the argument ''t'' often represents time (in seconds), and the function ''ƒ''(''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''.
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| When working with functions of time, ''ƒ''(''t'') is called the '''time domain''' representation of the signal, while ''F''(''s'') is called the '''s-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.
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| ==Relationship to other integral transforms==
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| If ''u''(''t'') is the [[Heaviside step function]], equal to zero when ''t'' is less than zero, to one-half when ''t'' equals zero, and to one when ''t'' is greater than zero, then the Laplace transform <math>\mathcal{L}</math> may be defined in terms of the two-sided Laplace transform by
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| :<math>\mathcal{L}\left\{f(t)\right\} = \mathcal{B}\left\{f(t) u(t)\right\}.</math>
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| On the other hand, we also have
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| :<math>\left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{L} f(t)\right\}(s)
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| + \left\{\mathcal{L} f(-t)\right\}(-s)</math>
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| so either version of the Laplace transform can be defined in terms of the other.
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| The [[Mellin transform]] may be defined in terms of the two-sided Laplace
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| transform by
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| :<math>\left\{\mathcal{M} f\right\}(s) = \left\{\mathcal{B} f(e^{-x})\right\}(s)</math>
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| and conversely we can get the two-sided transform from the Mellin transform by
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| :<math> \left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{M} f(-\ln x) \right\}(s).</math>
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| The Fourier transform may also be defined in terms of the two-sided Laplace
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| transform; here instead of having the same image with differing originals, we
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| have the same original but different images. We may define the Fourier transform
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| as
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| :<math>\mathcal{F}\left\{f(t)\right\} = F(s=i\omega) = F(\omega). </math>
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| Note that definitions of the Fourier transform differ, and in particular
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| :<math>\left\{\mathcal{F} f\right\}= F(s=i\omega) = \frac{1}{\sqrt{2\pi}}\left\{\mathcal{B} f\right\}(s)</math> | |
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| is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as
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| :<math>\left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{F} f\right\}(-is).</math>
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| 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.
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| The [[moment-generating function]] of a continuous [[probability density function]] ''ƒ''(''x'') can be expressed as <math>\left\{\mathcal{B} f\right\}(-s)</math>.
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| ==Properties==
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| It has basically the same properties of the unilateral transform with an important difference
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| {| class="wikitable"
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| |+ '''Properties of the unilateral Laplace transform'''
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| !
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| ! Time domain
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| ! unilateral-'s' domain
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| ! bilateral-'s' domain
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| |-
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| ! [[Derivative|Differentiation]]
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| | <math> f'(t) \ </math>
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| | <math> s F(s) - f(0) \ </math>
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| | <math> s F(s) \ </math>
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| |-
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| ! Second [[Derivative|Differentiation]]
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| | <math> f''(t) \ </math>
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| | <math> s^2 F(s) - s f(0) - f'(0) \ </math>
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| | <math> s^2 F(s) \ </math>
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| |}
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| To use the bilateral transform is equivalent to assume null initial conditions. Therefore it is more suitable than the unilateral for calculating transfer functions from the differential equations, or when looking for an easy particular solution.
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| ==Causality==
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| Bilateral transforms don't respect [[causality]]. They make sense when applied over generic functions but when working with functions of time (signals) unilateral transforms are preferred.
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| ==See also==
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| *[[Causal filter]]
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| *[[Acausal system]]
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| *[[Causal system]]
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| *[[Sinc filter]] - ideal sinc filter (aka rectangular filter) is acausal and has an infinite delay.
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| ==References==
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| *[[Wilbur R. LePage|LePage, Wilbur R.]], ''Complex Variables and the Laplace Transform for Engineers'', Dover Publications, 1980
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| * van der Pol, Balthasar, and Bremmer, H., ''Operational Calculus Based on the Two-Sided Laplace Integral'', Chelsea Pub. Co., 3rd edition, 1987
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| {{DEFAULTSORT:Two-Sided Laplace Transform}}
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| [[Category:Integral transforms]]
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