We have the bilateral Laplace transform:
where
And we also have the Fourier transform:
Fourier to Laplace
The two formula looks very similar and it is tempting to set
However, there is a caveat: only when the region of convergence (ROC) for
Laplace to Fourier
Conversely, we can start at the bilateral Laplace transform and try to get a Fourier transform. Since
In other words, Laplace transform can be understand as a Fourier transform of the original function multiplied by