I am particularly interested in 'why' it Works for ac theory. I say 'why' because I know 'how' it works - ie. how to do the transform.
I am puzzled that combinations of voltage waveforms, together with integrated and differentiated waveforms, can be combined, transformed to another space, manipulated and transformed back and that they successfully solve a circuit problem. The maths works, but why does the Physics? I was hoping the maths would help me understand the Physics.
On comments: I can see exp(-st) as an eigenfunction - it doesn't change with d/dt, its job seems to be to cause the transformed values to be finite.
The Gibbs distribution may be along the lines I was thinking - which is that perhaps that the transformed function is an encapsulation of the energy beneath the decaying exp(-st) voltage.
Stieltjes didn't take me anywhere towards an understanding.
"Laplace transform is closely related to the Fourier transform." Yes I understand that. Laplace just adds a real part to s and that causes a decay - the transformed function then occupies 2D in the W plane with both real and imaginary axes.
You have given me more to think about, for which I am grateful, but it doesn't seem to be leading me to a physical reason for it's success. Some of your answers are going to take me quite a while to digest, but I so grateful thank you took the time to give me for them.
After being on MathFlow the other day I was led to this which helped quite a bit:
and file itself: