# Two-sided Laplace transform

Two-sided Laplace transform

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 "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:$mathcal\left\{B\right\} left\left\{f\left(t\right) ight\right\} = F\left(s\right) = int_\left\{-infty\right\}^\left\{infty\right\} e^\left\{-st\right\} f\left(t\right) dt$There seems to be no generally accepted notation for the two-sided transform, the $mathcal\left\{B\right\}$ used here recalls "bilateral". The two-sided transformused by some authors is:$mathcal\left\{T\right\}left\left\{f\left(t\right) ight\right\} = smathcal\left\{B\right\}left\left\{f ight\right\} = sF\left(s\right) =s int_\left\{-infty\right\}^\left\{infty\right\} e^\left\{-st\right\} f\left(t\right) dt$

In science and engineering 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 $mathcal\left\{L\right\}$ may be defined in terms of the two-sided Laplace transform by:$mathcal\left\{L\right\}left\left\{f\left(t\right) ight\right\} = mathcal\left\{B\right\}left\left\{f\left(t\right) u\left(t\right) ight\right\}$On the other hand, we also have:$left\left\{mathcal\left\{B\right\} f ight\right\}\left(s\right) = left\left\{mathcal\left\{L\right\} f\left(t\right) ight\right\}\left(s\right) + left\left\{mathcal\left\{L\right\} f\left(-t\right) ight\right\}\left(-s\right)$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:$left\left\{mathcal\left\{M\right\} f ight\right\}\left(s\right) = left\left\{mathcal\left\{B\right\} f\left(e^\left\{-x\right\}\right) ight\right\}\left(s\right)$and conversely we can get the two-sided transform from the Mellin transform by:$left\left\{mathcal\left\{B\right\} f ight\right\}\left(s\right) = left\left\{mathcal\left\{M\right\} f\left(-ln x\right) ight\right\}\left(s\right)$

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:$mathcal\left\{F\right\}left\left\{f\left(t\right) ight\right\} = F\left(s=iomega\right) = F\left(omega\right)$Note that definitions of the Fourier transform differ, and in particular :$left\left\{mathcal\left\{F\right\} f ight\right\}= F\left(s=iomega\right) = frac\left\{1\right\}\left\{sqrt\left\{2pileft\left\{mathcal\left\{B\right\} f ight\right\}\left(s\right)$is often used instead.In terms of the Fourier transform, we may also obtain the two-sided Laplacetransform, as:$left\left\{mathcal\left\{B\right\} f ight\right\}\left(s\right) = left\left\{mathcal\left\{F\right\} f ight\right\}\left(-is\right)$

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

The moment-generating function of a continuous probability density function "f"("x") can be expressed as $left\left\{mathcal\left\{B\right\} f ight\right\}\left(-s\right)$.

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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Two-sided — may refer to:* Two sided Laplace transform, integral transform closely related to the Fourier transform, Mellin transform, and ordinary Laplace transform * Two sided ideal, a type of ideal in ring theory * Two sided markets, economic networks… …   Wikipedia

• Laplace transform — In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily soluble algebraic equation from an ordinary differential equation. It has many important… …   Wikipedia

• Z-transform — In mathematics and signal processing, the Z transform converts a discrete time domain signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. It is like a discrete equivalent of the Laplace… …   Wikipedia

• Mellin transform — In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used …   Wikipedia

• Integral transform — In mathematics, an integral transform is any transform T of the following form:: (Tf)(u) = int {t 1}^{t 2} K(t, u), f(t), dt.The input of this transform is a function f , and the output is another function Tf . An integral transform is a… …   Wikipedia

• Weierstrass transform — In mathematics, the Weierstrass transform [Ahmed I. Zayed, Handbook of Function and Generalized Function Transformations , Chapter 18. CRC Press, 1996.] of a function f : R rarr; R is the function F defined by:F(x)=frac{1}{sqrt{4piint {… …   Wikipedia

• Zweiseitige Laplace-Transformation — In der Mathematik, bezeichnet man mit der zweiseitigen Laplace Transformation eine Integraltransformation, die nahe verwandt zu der gewöhnlichen, zur Unterscheidung manchmal auch einseitig genannten, Laplace Transformation ist. Definition Für… …   Deutsch Wikipedia

• List of transforms — This is a list of transforms in mathematics.Integral transforms*Abel transform *Fourier transform **Short time Fourier transform *Hankel transform *Hartley transform *Hilbert transform **Hilbert Schmidt integral operator *Laplace transform… …   Wikipedia

• Convolution — For the usage in formal language theory, see Convolution (computer science). Convolution of two square pulses: the resulting waveform is a triangular pulse. One of the functions (in this case g) is first reflected about τ = 0 and then offset by t …   Wikipedia

• Fourier analysis — In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions. The… …   Wikipedia