We have the bilateral Laplace transform:

where is a complex frequency.

And we also have the Fourier transform:

Fourier to Laplace

The two formula looks very similar and it is tempting to set to get the Fourier transform from the Laplace transform.

However, there is a caveat: only when the region of convergence (ROC) for includes the imaginary axis, then exists and equals the Fourier transform .

Laplace to Fourier

Conversely, we can start at the bilateral Laplace transform and try to get a Fourier transform. Since , we have

In other words, Laplace transform can be understand as a Fourier transform of the original function multiplied by 1

Footnotes

  1. Laplace Transform Region of Convergence Explained (“THE best explanation I’ve seen”) - YouTube