$ X(s) $ s is a complex variable
$ X(s) = \int_{-\infty}^{\infty}x(s)e^{-st}dt $
The Laplace Transform $ X(s) $ evaluated on the imaginary axis $ X(j\omega) $ is equal to the F.T> at $ \omega $
So the F.T. is the restriction of the L.T. on the imaginary axis, $ s=j\omega $
$ X(s) = \int_{-\infty}^{\infty}x(s)e^{-st}dt = \int_{-\infty}^{\infty}x(s)e^{-(a+j\omega)t}dt = \int_{-\infty}^{\infty}x(s)e^{-at}+e^{j\omega t}dt = F(x(t)e^{-t}) $
Where a is the real part of s.
