Two-sided Laplace transform

In mathematics , the two-sided Laplace transformation is an integral transformation that is closely related to the usual Laplace transformation, which is sometimes called one-sided to distinguish it.

definition

For a real or complex valued function of a real variable , the two-sided Laplace transform for all complex numbers is through the integral ${\displaystyle f(t)}$ ${\displaystyle t}$ ${\displaystyle s}$

${\displaystyle {\mathcal {B}}\left\{f(t)\right\}=F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)dt}$

Are defined.

The difference to the usual Laplace transform is the integration of up instead of over . ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle (0,\infty )}$

In systems theory , the two-sided Laplace transformation, in contrast to the usual one-sided Laplace transformation, only plays a subordinate role. The reason is that in physics and technology only causal systems that occur can be described with the one-sided Laplace transformation. In the theoretical analysis of non-causal systems, these are systems that show an effect before the triggering cause, the two-sided Laplace transform is to be used, which, depending on the function , shows poor convergence behavior. For causal systems, the result of the two-sided Laplace transform is identical to the usual one-sided Laplace transform. The two-sided Laplace transform also occurs in probability theory for moment-generating functions . ${\displaystyle f(t)}$ ${\displaystyle t\to -\infty }$

context

With the Heaviside function , the two-sided and the one-sided Laplace transformation can be set in the following context: ${\displaystyle \epsilon (t)}$ ${\displaystyle {\mathcal {L}}\left\{\cdot \right\}}$

${\displaystyle {\mathcal {L}}\left\{f(t)\right\}={\mathcal {B}}\left\{f(t)\epsilon (t)\right\}}$

Equally important, the following relationship exists between the two transformations:

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

The following relationship exists with the Mellin transformation : ${\displaystyle {\mathcal {M}}\left\{\cdot \right\}}$

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

and the inverse relationship:

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

literature

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