So I’m looking for some applications that are relatively easy to understand, ideally in like CS or physics, but anything works. I’ve seen it be talked about in discussions of oscillations (I have a pretty good basic understanding of pendulum motion, for example, but I don’t understand how complex numbers come in), or in AC currents (I have like 0 background in electricity and magnetism apart from basic high school physics with super basic DC systems), but I can never understand how we use complex numbers for that…
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The pendulum is a great example of a physical system governed by second-order differential equations of motion. If you assume small-angle motion, you can approximate the EOM with a linear differential equation. Then you use an awesome tool called the Laplace transform (a more general form of the Fourier transform that someone else mentions) to transform the linear differential equation into a polynomial algebraic equation. This is where complex numbers earn their keep bigtime.
The pendulum leads to a quadratic equation. You can tell all you need to know about the pendulum’s behavior by finding the roots of that quadratic equation. There will always be two roots (because all quadratic equations have two roots). If the pendulum is undamped, the roots are pure imaginary. If there is energy dissipation, the roots are complex conjugates, with negative real parts. If something is putting energy in (“negative damping”), then the roots are complex conjugates with positive real parts.
You can transform back into differential equation land, thus finding the time-domain solution to the differential equation, but you can pick off a lot of useful information just by where the roots are in the complex plane. Does any root have a positive real part? Then the system is unstable. How far are the roots from the origin? That gives you the natural frequency of the system. How far are the roots from the imaginary axis? That tells you how well damped the motion is.
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