Editing Two-sided Laplace transform (section)

==Relationship to other integral transforms==
If ''u'' is the [[Heaviside step function]], equal to zero when its argument is less than zero, to one-half when its argument equals zero, and to one when its argument is greater than zero, then the Laplace transform <math>\mathcal{L}</math> may be defined in terms of the two-sided Laplace transform by
:<math>\mathcal{L}\{f\} = \mathcal{B}\{f u\}.</math>

On the other hand, we also have
:<math>\mathcal{B}\{f\} = \mathcal{L}\{f\} + \mathcal{L}\{f\circ m\}\circ m,</math>
where <math>m:\mathbb{R}\to\mathbb{R}</math> is the function that multiplies by minus one (<math>m(x) = -x</math>), 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
:<math>\mathcal{M}\{f\} = \mathcal{B}\{f \circ {\exp} \circ m\},</math>
with <math>m</math> as above, and conversely we can get the two-sided transform from the Mellin transform by
:<math>\mathcal{B}\{f\} = \mathcal{M}\{f\circ m \circ \log \}.</math>

The [[moment-generating function]] of a continuous [[probability density function]] ''&fnof;''(''x'') can be expressed as <math>\mathcal{B}\{f\}(-s)</math>.