* The Wikipedia example of using Fourier series to estimate periodic waves
Since the last blog, I’ve been learning about Fourier Series. In his previous lectures about solving ODEs, Prof. Mattuck warns his students to pay extra attention to trigonometric functions. Of course he means that it is easy to make mistakes with integrating/differentiating sinusoidal functions (sin and cos), yet this week’s lecture videos prove me wrong.
As hinted in the first sentence, the true reason is because of Fourier series. The term “series” should be familiar to any AP Calc BC test takers because there is Maclaurin/Taylor series. Many may also call them Taylor expansions, and the idea of using a series notation to re-write a function is the same for Fourier series.
* the above and below shows a general form of Maclaurin series and a form of Fourier series, respectively.
The purpose of using a Taylor/Maclaurin expansion is so that in some cases, it is very convenient to estimate the value of the function. For Fourier series, although it is true that we can use such method to get an estimate, but the reason Fourier series is introduced in differential equations is so that we can now figure out all the problems with periodic functions (with the assumption that periodic functions are all express-able by a Fourier series). Although the exact process of converting periodic functions to Fourier series is yet-to-learn for me, I found this idea really interesting. In the past, I’ve always wondered what periodic functions are other than trig functions. When I read the book One, Two, Three…Infinity, I came across a beautiful thing called Weierstrass function. At that time, I had just stepped into the world of calculus; and reading about such a continuous-everywhere-but-differentiable-nowhere function blew my mind. (Also, the name Weierstrass was just so funky to read out loud!) It is written in Fourier series and I wondered what that was. To a certain degree, my curiosity about Weierstrass function is what propelled me through the difficult math learning journey so far. Beginning to unveil the deep core of that wonder of mine is nothing but exciting.
P.S. By all means, here is the weird Weierstrass function:
Here’s an animated version of how Fourier series can behave: