Two-sided Laplace transform

In mathematics, one designated by the two-sided Laplacian an integral transformation, Laplace transform is closely related to the ordinary for distinguishing said one side sometimes.

Definition

For a real - or complex-valued function f ( t) of a real variable t is the two-sided Laplace transform for all real numbers s by the integral

Defined.

The difference from the normal Laplace transform is the integration of to take place over.

In system theory, the two-sided Laplace transform plays, as opposed to the usual one-sided Laplace transform, only a subordinate role. The reason is that it is possible in physics and technology exclusively occurring causal systems described by the one-sided Laplace transform. In the theoretical analysis of non- causal systems, these are systems that show an effect against the underlying cause, the two-sided Laplacian transform is to be used, which having a function of the function of poor convergence behavior. For causal systems, the result of the two-sided Laplacian transformation is identical to the ordinary one-sided Laplacian. The two-sided Laplace transform It also occurs in probability theory with moment generating functions.

Context

With the Heaviside function can be set in the following context, the two-sided with the one-sided Laplace transform:

These equivalent between the two transformations the following relationship:

Using the Mellin transformation, the following relationship exists:

And the inverse relationship: