At Last! (warning: pure nerdiness)

Fourier series come up all the time in physics and engineering. All, all, all the time. And yet, I somehow managed to get this far by seeing "Fourier series" and thinking "Oh, shoot, I hate this part. Now we get lots and lots and lots of functions, and something about sine waves, and converting between stuff, and if wait quietly, it will go away."

Turns out, the idea of a Fourier series is incredibly straightforward. Hopefully I knew it once and just forgot...

In a nutshell-- (What, you thought I wasn't going to go all-out nerd on you? Dream on!)

Well, in a nutshell, look at Fig. 1 from Kovesi's "Symmetry and Asymmetry from the Local Phase," which "shows the Fourier series representation of both a square wave and a triangular wave."

Notice that if you add the values of all of the dashed curves at any given point, you get the value of solid line at that point. Thus in each graph, the collection of dashed curves is the Fourier series which approximates that solid curve. Because that's all you're doing if you're using a Fourier series: adding together many (usually well-understood and well-behaved) functions--a "series"of functions, if you would--in order to approximate a single (often hairy and annoying) function.

And that is an almost perfectly backwards way of understanding a Fourier series (starting from a very specific case, and working back to the general principle, instead of vice-versa), but now I get it. So there. At last!