- Two-sided Laplace transform
In

mathematics , the**two-sided Laplace transform**or**bilateral Laplace transform**is anintegral transform closely related to theFourier transform , theMellin transform , and the ordinary or one-sidedLaplace 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:$mathcal\{B\}\; left\{f(t)\; ight\}\; =\; F(s)\; =\; int\_\{-infty\}^\{infty\}\; e^\{-st\}\; f(t)\; dt$There seems to be no generally accepted notation for the two-sided transform, the $mathcal\{B\}$ used here recalls "bilateral". The two-sided transformused by some authors is:$mathcal\{T\}left\{f(t)\; ight\}\; =\; smathcal\{B\}left\{f\; ight\}\; =\; sF(s)\; =s\; int\_\{-infty\}^\{infty\}\; e^\{-st\}\; f(t)\; dt$In

science andengineering applications, the argument "t" often represents time (in seconds), and the function "f"("t") often represents a signal or waveform that varies with time. In these cases, "f"("t") is called the**time domain**representation of the signal, while "F"("s") is called the**frequency 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.**Relationship to other integral transforms**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 $mathcal\{L\}$ may be defined in terms of the two-sided Laplace transform by:$mathcal\{L\}left\{f(t)\; ight\}\; =\; mathcal\{B\}left\{f(t)\; u(t)\; ight\}$On the other hand, we also have:$left\{mathcal\{B\}\; f\; ight\}(s)\; =\; left\{mathcal\{L\}\; f(t)\; ight\}(s)\; +\; left\{mathcal\{L\}\; f(-t)\; ight\}(-s)$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:$left\{mathcal\{M\}\; f\; ight\}(s)\; =\; left\{mathcal\{B\}\; f(e^\{-x\})\; ight\}(s)$and conversely we can get the two-sided transform from the Mellin transform by:$left\{mathcal\{B\}\; f\; ight\}(s)\; =\; left\{mathcal\{M\}\; f(-ln\; x)\; ight\}(s)$

The Fourier transform may also be defined in terms of the two-sided Laplacetransform; here instead of having the same image with differing originals, wehave the same original but different images. We may define the Fourier transform as:$mathcal\{F\}left\{f(t)\; ight\}\; =\; F(s=iomega)\; =\; F(omega)$Note that definitions of the Fourier transform differ, and in particular :$left\{mathcal\{F\}\; f\; ight\}=\; F(s=iomega)\; =\; frac\{1\}\{sqrt\{2pileft\{mathcal\{B\}\; f\; ight\}(s)$is often used instead.In terms of the Fourier transform, we may also obtain the two-sided Laplacetransform, as:$left\{mathcal\{B\}\; f\; ight\}(s)\; =\; left\{mathcal\{F\}\; f\; ight\}(-is)$

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

The

moment-generating function of a continuousprobability density function "f"("x") can be expressed as $left\{mathcal\{B\}\; f\; ight\}(-s)$.**References***LePage, Wilbur R., "Complex Variables and the Laplace Transform for Engineers", Dover Publications, 1980

* 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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