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

$\backslash mathcal\{B\}\; \backslash left\backslash \{f(t)\backslash right\backslash \}\; =\; F(s)\; =$

\int_{-\infty}^{\infty} e^{-st} f(t) dt There seems to be no generally accepted notation for the two-sided transform, the $\backslash mathcal\{B\}$ used here recalls "bilateral". The two-sided transform used by some authors is

$\backslash mathcal\{T\}\backslash left\backslash \{f(t)\backslash right\backslash \}\; =\; s\backslash mathcal\{B\}\backslash left\backslash \{f\backslash right\backslash \}\; =\; sF(s)\; =$

s \int_{-\infty}^{\infty} e^{-st} f(t) 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

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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 $\backslash mathcal\{L\}$ may be defined in terms of the two-sided Laplace transform by

$\backslash mathcal\{L\}\backslash left\backslash \{f(t)\backslash right\backslash \}\; =\; \backslash mathcal\{B\}\backslash left\backslash \{f(t)\; u(t)\backslash right\backslash \}$

On the other hand, we also have

$\backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{L\}\; f(t)\backslash right\backslash \}(s)$

+ \left\{\mathcal{L} f(-t)\right\}(-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

$\backslash left\backslash \{\backslash mathcal\{M\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{B\}\; f(e^\{-x\})\backslash right\backslash \}(s)$

and conversely we can get the two-sided transform from the Mellin transform by

$\backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{M\}\; f(-\backslash ln\; x)\; \backslash right\backslash \}(s)$

The Fourier transform may also be defined in terms of the two-sided Laplace transform; here instead of having the same image with differing originals, we have the same original but different images. We may define the Fourier transform as

$\backslash mathcal\{F\}\backslash left\backslash \{f(t)\backslash right\backslash \}\; =\; F(s=i\backslash omega)\; =\; F(\backslash omega)$

Note that definitions of the Fourier transform differ, and in particular

$\backslash left\backslash \{\backslash mathcal\{F\}\; f\backslash right\backslash \}=\; F(s=i\backslash omega)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(s)$

is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as

$\backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{F\}\; f\backslash right\backslash \}(-is)$

The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip which may not include the real axis.

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

References

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• LePage, Wilbur R., Complex Variables and the Laplace Transform for Engineers, Dover Publications, 1980
• van der Pol, Balth., and Bremmer, H., Operational Calculus based on the Two-sided Laplace Transform, Cambridge University Press, 1964

integral transforms