In the Laplace Transform, we perform:
$$F(s) = \int_{0}^{\infty}f(t)e^{-st}dt$$
With some interpretations of $s$ being that $s = \sigma + i \omega$. It is natural to consider $\omega$ as the frequency of a sine wave, but what about $\sigma$? Is it phase? If so, with respect to what?
I know that in engineering, a second order system is represented in the $s$ domain and the location of the poles of that system relate to the damping coefficient of the system. In that case, the $s$ coordinates of the poles make sense as a damping term. However, the Laplace Transform is not defined only at the poles of rational functions. So here are two questions, one of which is a special case:
In terms of the engineering case I mentioned above, what does $\sigma$ represent in terms of the "frequency domain" representation of a physical system's mathematical model?
More generally, what is the interpretation of that real part?