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(x) is a real or complex valued function of the real variable x defined for all real numbers, then the two-sided Laplace transform is defined by the integral

$\left\{\mathcal{B} f\right\}(s) = \varphi(s) = \int_{-\infty}^{\infty} e^{-sx} f(x) dx$

There seems to be no generally accepted notation for the two-sided transform, the $\mathcal{B}$ used here recalls "bilateral". The two-sided transform used by some authors is

$\left\{\mathcal{T} f\right\}(s) = s\left\{\mathcal{B} f\right\}(s) = s \int_{-\infty}^{\infty} e^{-sx} f(x) dx$

Relationship to other integral transforms

If u(x) is the Heaviside step function, equal to zero when x is less than zero, to one-half when x equals zero, and to one when x is greater than zero, then the Laplace transform may be defined in terms of the two-sided Laplace transform by

$\left\{\mathcal{L} f\right\}(s) = \left\{\mathcal{B} uf\right\}(s)$

On the other hand, we also have

$\left\{\mathcal{B} f\right\}(s) = \left\{\mathcal{L} f(x)\right\}(s) + \left\{\mathcal{L} f(-x)\right\}(-s)$

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

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

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

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

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

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

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

$\left\{\mathcal{F} f\right\}(is) = \frac{1}{\sqrt{2\pi}}\left\{\mathcal{B} f\right\}(s)$

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

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

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.

References

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

Two-sided Laplace transform

Relationship to other integral transforms

References

See also: Two-sided Laplace transform, Fourier transform, Heaviside step function, Integral transform, Laplace transform, Mathematics, Mellin transform