Consider the Laplace transform of an RC filter. For those who can't immediately summon it, refer equation (46) at this link: http://web.mit.edu/2.151/www/Handouts/FreqDomain.pdf for a refresher.
In the same file, equations (47) and (48) represent the frequency response function of the network. Now I am sure that I, like many others who will read this question, could have obtained the frequency response just by using time domain analysis. Inserting $σ = 0$ is just one of the many conveniences Laplace transforms affords us.
But I would like to know why that is so. Why replacing $s$ with $jω$ yields the frequency response? More than the answer to this question I would like to have an intuitive understanding of Laplace transforms and the s-domain.
I could figure out many properties of
addition and the
numbers domain before I was told about those properties. For instance, I knew about
commutative properties of addition before they were formally introduced; i.e. intuition preceded formal definition. But I could have never figured out, just by learning Laplace transforms, as to why replacing $s$ with $jω$ yields what it yields.
I would be grateful if anyone has a deeply intuitive understanding of the Laplace transform and could answer my question through that understanding. I just watched this video http://www.youtube.com/watch?v=hqOboV2jgVo which explains how we arrive at Laplace transforms. Brian Douglas has posted some really cool videos on YouTube regarding the same topic. But I still feel like I am grappling in the dark. I guess an intuition, if indeed there's one for humans, will automatically answer many basic questions about Laplace transforms including this one.