I'm trying to understand the Laplace Transform (LT) on an intuitive level, if possible. In particular I'm struggling to understand what the s-domain represents.
I get the fact that the LT is defined similarly to the Fourier Transform (FT) except that the exponential has an arbitrary complex exponent, instead of a purely imaginary one. So when the real component of this complex number is zero, we just get the FT.
Now, the FT is a way of decomposing the original function, and is essentially a 2D "slice" of the LT taken at the imaginary axis. If I understand it correctly, the other slices parallel to this are just different ways of decomposing the same original function. But is that only true for slices that are parallel to the imaginary axis? For example does the slice corresponding to the real axis represent the original function, decomposed into purely real exponentials?
(If this is the wrong way of thinking about it and not really answerable, then please feel free to answer the question you think I should have asked).