Two-sided Laplace Transform - Relationship To Other Integral Transforms

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 may be defined in terms of the two-sided Laplace transform by

On the other hand, we also have

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

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

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

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

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

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

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 ƒ(x) can be expressed as .

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