What does s=jω actually mean in terms of the complex plane and Laplace transforms?

Question

I was trying to solve a problem on RC circuits. The current source was of the form $\cos (\omega t)$ which transforms in the manner of Laplace to $\frac{s}{s^2+\omega^2}$. I thought I’d use the complex reactance for the capacitor which is $1/j \omega C$ and so I should replace $s$ in the above term with $j \omega$. But that would make the denominator of the term infinite. What am I missing?

Answer 1

In general the complex variable s is equal to a+jw, where a is the exponential damping portion of the complex frequency. When s is equal to only jw, it means you are considering a small portion of the complex response related to only the imaginary axis in the s plane, which in this case would be a purely reactive circuit (i.e. no resitance). This leads to the result that your current source would put out infinite current with no resistance is the circuit.