How do the Laplace s-domain and the complex frequency domain differ? In systems theory and signal processing, we often transform expressions based in the Laplace $s$-domain into the complex frequency domain with $j\omega$ (engineering notation for the angular frequency on the imaginary axis): $s \leftrightarrow j\omega$.
I have always been told I can easily transform expressions between the two, but that I shouldn't simply equate them. So far, I always simply considered the Laplace $s$ as a 'sort of frequency or pulsation'. Is this correct?
What exactly is the difference in meaning between these two domains?
PS: Feel free to (re)tag, as I am not familiar with this particular forum's tags.
 A: The Laplace transform turns linear differential equations into algebraic ones.  Multiplication by $s$ is the operation corresponding to differentiation wrt to $t$ in the other domain.  Maybe you should think of it as an operator, not a quantity like a generalized frequency.
In an AC circuit, there's a power source with sinusoidal voltage or current, and other elements that are proportional to it, its derivative, or its integral.  Resistors, inductors, capacitors.  In this case, $s=jw$ because the source is sinusoidal by design, and the derivative of $e^{jwt}$ is $jw e^{jwt}$.  In other applications, we can't assume sinusoids everywhere, and $s$ has nothing to do with frequencies. 
You might like Wilbur LePage's book, published by Dover.
A: In simple terms, the Fourier transform $F(j\omega)$ is defined through the imaginary axis, while the Laplace transform $F(s)$, where $s=j\omega+\sigma$ is defined in the whole complex plane.
The Fourier transform only makes real sense when is used on periodic signals, while the Laplace transform is used with signals with initial value ($f(0^+)=\lim_{s\rightarrow \infty }sF(s)$) and a final one ($f(\infty)=\lim_{s\rightarrow 0 }sF(s)$), mostly defined in terms of a Dirac delta (or impulse) or a step function.
The above means that the Laplace transform is the proper tool to algebraically handle systems of differential equations with initial conditions and input signals based on steps or impulses.
