Complex signals can represent circuits fairly accurately. Complex signals consist of a real component along with an imaginary component. The imaginary component is represented by the letter j if your an engineer, a letter i for the rest of the mathematics literal world.
j=$ \sqrt{-1} $
$ j^2=-1 $
An important conversion:
$ x(t)=e^{j\omega t} $
$ e^{j\omega t}= cos\omega t + jsin\omega t $
An example of a complex signal/system would be x = 10 + 12j
Complex signals can do a fairly good job of describing systems
involving circuits and springs! As you can see above in the "Important
conversion" $ e^{j\omega t} $ is essentially a complex
number that represents typically oscillating mathematical symbols.
What we care about is that by altering the $ \omega $ we can represent periodic oscillating systems as well as damped and undamped, but we really care about the periodic ones.
We can test to see if our function\signal is periodic by $ {\omega/ 2\Pi = rational number!} $
