Laplace and Fourier transforms

It was a quite long ago when I first met Laplace transform. Later, I also learned Fourier (most famous transform in EE in general ^^) and z-transform. However, it took me quite while to understand them and actually use them. I am sure I don't understand them all of their properties and usefulness.

I basically understood that Fourier transform was a subset of Laplace transform. That's all I understood in my undergrad. Now? Not much...actually.
However, last year I had to solve one of equations using Laplace tranform and had a chance to re-think about its property.
The followings are what I gave thought about Lapalce and Fourier transform.

1. Laplace is a linear tranform (so is Fourier transform). Thus, it should be applied to linear systems only.

2. Frequency reponse of a system is basically the response seen from jw axis out of 3D figure in s-domain.

3. jw-axis in S-domain is defined as when sigma = 0, which means all the transient responses die out so that only response we can get out of the frequency response is "steady-state" response. Therefore, you don't expect any transient response while you solve a question in frequency response. This is simple, but often confusing. For example, you want to calculate noise response of a system. Let's assume that only noise source is a resistor - thermal noise. From typical text book says, a resistor's power spectral density (PSD) is equal to 4KTR [Vrms^2]. If you analyze a circuit with this equation, you are intersted in noise analysis in steady-state!!! Noise power you calculate in the end only counts "steady-state" noise power. You never see how noise power changes over time because we didn't consider transient response! In conclusion, the in frequency domain (i.e. 4KTR in this example) is valid only when Linear time-invariant steady-state case! (when sigma = 0 in Laplace domain)

* I think this is a good insight I have overlooked for long time!

4. Many other properties are stated in many books....so...this is it!!! This is all I want to mention!