Hi,

To give you a first answer... First: You should be perfectly fine here in Complex Anaylysis. Hopefully you get a fully qualified answer, otherwise if you are more interested in the physical application I would tend to

http://www.physicsforums.com.

I am sort of irritated by your third point. So some words about your issue. You will know Fourier transform. The idea is to understand a function in the original domain better in another domain. Say, you have a pure sine function with frequency f which is in the time domain in y-t-coordinates an endless curve, going from -1 to 1. You couldn't print it on paper since it is endless. After beeing transformed it is not only printable on a piece of paper, it is just one line, you have for example the y-axis as the amplitude and there is just one point in the frequency domain, exactly at f.

What you learn from this is how the energy is distributed over the frequency domain, for example in which modes of vibration the main energy is concentrated.

Question: When does this work and when does it not work? The main point is that such an analysis makes sence if you have a basic idea about the signal/function you would like to understand better. Example: You have an measurement which seems to be linear. Would you think about Fourier transforming this one time measurement? For sure not. Linear regression gives you information, hidden in the data.

Now let us move to Laplace transforms. Say you have a black box and you are interested what happens if you are activating it. Before, there was nothing. You plug it in and now you get a signal. The common situation is that you will have after some time T another signal than in the very beginning, think of an capicitor to be loaded first. It is absolutely senseless to describe a non repeating exponential function to be linear or to be constructed of sine-functions. So, why not to understand the function as an exponential function? This is the only natural way. Say, the function is simply exp(-s * t). Again, in the time domain you have an endless plot without any deeper information. In the "frequency" (s) domain you just note the value s, and you are happy!

Next step: We have again that black box. We plug it in and want to see what happens from t=0. We expect in the beginning probably an exponential behaviour, then it seems to fluctuate. Fourier transform would be misleading. You miss the energy in the beginning - and here you seem to have something completely misunderstood. You simply use as the kernel of the integral a more general function exp(j * a * t) * exp(b * t)=exp([j a + b]*t)=exp(s t)

After this step you will get new information in the complex s domain! Just adding, to give you an overview about integral kernels, please have a look here:

http://en.wikipedia.org/wiki/Integral_transform
Best wishes,

Jens