Impedance in the Frequency Domain

Was looking through Network Analysis for this week's test. I recall Biji sir having substituted sL and 1/sC as impedance in the frequency domain being given the initial conditions. I had wondered HOW that was feasible and asked around but nobody at the time could give a straight answer (As for asking sir himself the first and last time I did so ended in disaster). Having caught up a bit I have come up with two answers:

1) sL and 1/sC are multiplied with the respective current functions in the Laplace transformed KVL equations. So it is safe to assume that sL and 1/sC are indeed the impedance of the inductor and capacitor and write the KVL quation in the s-domain directly. (Ref. Network Analysis and Synthesis, Franklin F. Kuo)

2) (WARNING: Made this one up myself!) What we refer as s is simply a complex frequency variable of the form

s = σ + jω

And since jωL and 1/jωC are the impedences in polar form the assumption is valid. However there is still the question of what happened to σ. Any guesses?