[hamiltonkibbe]

For a little more intuition about why you can do that replacement: laplace is a representation of a system as a sum of sinusoids * exponentials, which are your two axis in the laplace plane. Frequency on the iw axis and exponential on the a axis. If you think of that replacement as s = iw + a | a = 0, you'll see the exponential terms go away and you're left with just the sinusoidal parts:

  f(t) * e^(iw + 0)t 
  = f(t) * e^(iwt) * e(at) 
  = f(t) * e(iwt) * e^(0t) 
  = f(t) * e(iwt) 

integrated over time, which is your fourier transform subject to the condition above. It's just the laplace transform along the Y axis, or, the frequency response at steady state when not growing/decaying exponentially.