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The periodic square wave $$ f(x) = \cases{ 1 \text{ if } 0 \le x \le \pi \\ 0 \text { if } -\pi \le x < 0} $$ seems easy enough to work with. Why transform it into a series of sines and cosines?

I'm writing up take-home bonus problems for my Calculus II class. I know one of my students is interested in Fourier series and it's relevant to many others, I'm sure, so one of the problems is a easy Fourier expansion of the square wave. I'd like to include in the introduction why someone might turn a seemingly-simple function like this into an infinite sum of trig functions.

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All modern digital audio uses the Fourier Transforms extensively, so knowing a square wave's Fourier series can be very helpful to understanding why, say, a hi or lo-pass filter changes its shape in various ways. learnabout-electronics.org/ac_theory/filters84.php –  RBarryYoung Mar 30 at 18:21

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up vote 10 down vote accepted

From the applied side, one could say that the motivation lies in trying to solve some partial differential equations like diffusion equations or wave equations.

For example, the diffusion of heat along a rod can be modeled as $$u_t = c^2u_{xx} \quad (*)$$ where $u(x,t)$ is the temperature at position $x$ of the rod at time $t$. We are given the initial distribution of heat: that is, at $t=0$, $u(x,0) = f(x)$. We wish to find a solution to $(*)$.

A common way to solve this involves writing $u$ as a (Fourier) series. In particular, if $f(x)$ can be written as $\sum_{k=1}^\infty a_k \sin(kx)$, then one has that $$u(x,t) = \sum_{k=1}^\infty a_k\sin(kx)e^{-k^2t}$$ is indeed the solution to $(*)$. Notice that even when $f(x)$ is as simple as the one you've given, $(*)$ is still hard to solve without use to Fourier series.

(Disclaimer: of course, technically I should be careful about boundary conditions of the differential equations and about the convergence of the sequence. I vastly simplified stuff since we are only trying to have a motivation.)

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I'm a physicist and I approve this answer! The square wave can for example be seen as an initial heat-distribution on a rod that you hit with some kind of heating-element with a finite width (and and homogeneous temperature). This finite width would coincide with the width of the square wave which serves as an initial condition for the temperature distribution. Now as roninpro points out in his comment the square wave (together with his friends the sawtooth et al.) also plays a role in electronics and signal processing. (read part 2 for a fair warning) ... –  Nick Mar 30 at 11:21
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(the fair warining), Be aware whenever you use a Fourier-series to solve practical problems! You can only describe PERIODIC processes, so if you'd study a transiënt proces (like the charging of the capacitor), then the Fourier-series would only be able to describe the equilibrium-regime (since you made a periodic extension as Avitus points out). –  Nick Mar 30 at 11:23
    
... and that's why God invented Fourier transforms ;) –  Neal Mar 30 at 12:56
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@Nick Ën̈g̈l̈ïs̈ḧ ïs̈ n̈öẗ G̈ër̈m̈än̈, ïẗ d̈öës̈ n̈öẗ üs̈ë ẗḧë d̈ïäër̈ës̈ïs̈. (also, the Georgia font has hopelessly broken combining characters) –  kinokijuf Mar 30 at 16:49
    
@kinokijuf, +1 and now I'll never forget :). –  Nick Apr 1 at 8:05

There are certainly many places within mathematics where the Fourier series / transform would be useful. I'd also like to mention that it's a huge component of many practical areas; one in particular is digital signal processing. The square wave you mentioned can literally be heard as a sound played out over time (i.e. signal represented in the time domain). What if you want to store that sound on a computer? And can you do it without using too much space? Computing the Fourier series / transform for the square wave lets you break it down into a set of frequencies (i.e. signal represented in the frequency domain). This leads to interesting ways to store and compress audio information. (A very simple example: viewing a signal in the frequency domain makes it easy to chop off frequencies inaudible to the human ear, reducing file size possibly at the cost of quality.)

You might be interested in:

Shannon-Nyquist Sampling Theorem

MP3 Encoding

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Here are two applications of such a development. In order to simplify matters I consider the following odd periodic function: $$f(x):=\cases{-1\quad&$(-\pi<x<0)$\cr 1\quad&$(0<x<\pi)$\cr f(x+2\pi)\quad&(all $x$) .\cr}$$ This $f$ can be developed into a sine series of the form $\sum_{k=1}^\infty b_k\>\sin(k t)$ with $$b_k={2\over\pi}\int_0^\pi f(t)\>\sin(kt)\ dt=\cases{{4\over\mathstrut \pi k}\quad&($k$ odd)\cr 0&($k$ even) ,\cr}$$ and is represented by this series at all points of continuity. In particular one has $$1=f\left({\pi\over2}\right)={4\over\pi}\sum_{k\geq 1\ {\rm odd}}^\infty{\sin(k\pi/2 )\over k}={4\over\pi}\left(1-{1\over3}+{1\over 5}-{1\over7}+\ldots\right)\ ,$$ from which we immediately get the famous $$1-{1\over3}+{1\over 5}-{1\over7}+\ldots={\pi\over 4}\ .$$ When Parseval's formula $${1\over4}|a_0|^2+{1\over2}\sum_{k=1}^\infty\bigl(|a_k|^2+|b_k|^2\bigr)={1\over 2\pi}\int_{-\pi}^\pi |f(t)|^2\ dt\tag{1}$$ is available we can obtain a more spectacular result. In our case the formula $(1)$ gives $${1\over2}\sum_{k\geq 1\ {\rm odd}}^\infty{16\over\pi^2 k^2}={1\over 2\pi}\cdot 2\pi=1\ ,$$ so that $$\sum_{k\geq 1\ {\rm odd}}^\infty{1\over k^2}={\pi^2\over 8}\ .$$ It follows that $$\zeta(2):=\sum_{k=1}^\infty{1\over k^2}=\sum_{k\geq 1\ {\rm even}}^\infty{1\over k^2}+\sum_{k\geq 1\ {\rm odd}}^\infty{1\over k^2}={1\over4}\zeta(2)+{\pi^2\over 8}\ ,$$ which finally gives $\ {\displaystyle\zeta(2)={\pi^2\over 6}}$.

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A $caveat$: the given function is not periodic, so no Fourier series can be directly computed. One introduces a periodic extension, let us say $\hat{f}$, of $f$ and to applies the Fourier series theory to it, instead. If we consider the function (or "signal")

$$f(x)=1,~~x\in[0,\pi]$$

then even / odd $2\pi$-periodic square wave extensions are available. Now, given the chosen periodic extension $\hat{f}$ of $f$ one considers its Fourier series because the series itself is just a linear combination of very simple periodic functions, and the function $\hat{f}$ is fully characterized by the set of "numbers" called Fourier coefficients, and its period.

This remarkable property becomes even more important when dealing with (arbitrarily) complex periodic functions. Considering that square waves are very common in engineering, the above simplification becomes relevant. As in most applications one can consider $de facto$ a finite subset of all Fourier coefficients of a given function, the gain in manipulating Fourier series becomes even more evident.

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why the downvote? –  Avitus Mar 30 at 15:01
    
Not sure. I did adjust my function though (I was sloppy with the "otherwise"). –  D. Weathers Mar 30 at 15:03

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