If you ask most people to explain the Fourier series they will tell you how you can decompose any particular wave into a sum of sine waves.  We’ve used that explanation before ourselves, and it is not incorrect.  In fact, it is how Fourier first worked out his famous series.  However, it is only part of the story and master video maker [3Blue1Brown] explains the story in his usual entertaining and informative way.  You can see the video below. 

Paradoxically, [3Blue1Brown] asserts that it is easier to understand the series by thinking of functions with complex number outputs producing rotating vectors in a two-dimensional space.  If you watch the video, you’ll see it is an easier way to work it out and it also lets you draw very cool pictures. 

Of course, our old friend the sine wave is just a special case where the imaginary part of the complex number is zero.  That makes for boring drawings, but you can imagine that if you understand the complex number situation, dealing with real numbers is easy. 

Another item in the video is the history of these equations.  Turns out Fourier wasn’t trying to solve anything having to do with waves.  He was working on the heat equation.  This video is part of a series on the heat equation, but it does just fine on its own. 

This topic comes up pretty regularly in part because it has so many applications from filtering to medical imaging.  If you want another visualization of how little circles make waves and not just pictures, we’ve looked at that before, too.