> The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors (decaying or rising) AND its sinusoidal components.
I want to clarify something a tiny bit misleading about this. In general complex exponentials are not orthogonal w.r.t. the relevant inner product. You can't really think of them as independent components that compose a function.
If you think of a function as the impulse response of a linear time invariant system, then the laplace transform of that function tells you the result of an experiment where you drive the system with an exponentially damped sinusoid. This is why the poles of the transformed impulse response tell you about the stability of the system: those are the inputs that cause the system to explode!
I want to clarify something a tiny bit misleading about this. In general complex exponentials are not orthogonal w.r.t. the relevant inner product. You can't really think of them as independent components that compose a function.
If you think of a function as the impulse response of a linear time invariant system, then the laplace transform of that function tells you the result of an experiment where you drive the system with an exponentially damped sinusoid. This is why the poles of the transformed impulse response tell you about the stability of the system: those are the inputs that cause the system to explode!