## Section10.13Fourier Series Example

Let's consider an example. Suppose $$f(x)$$ describes a square wave, so that

$$f(x) = \Theta(\frac{L}{2}-x)= \begin{cases} 1 \amp (0\le x\lt \frac{L}{2}) \\ 0 \amp (\frac{L}{2}\lt x\le L) \end{cases}\tag{10.13.1}$$

(the value of $$f$$ at the single point $$x=L/2$$ doesn't matter). According to the results of the previous sections, we have

$$f(x) = \frac12 a_0 + \sum_{n=1}^\infty a_n \cos\left(\frac{2\pi n x}{L}\right) + \sum_{n=1}^\infty b_n \sin\left(\frac{2\pi n x}{L}\right)\tag{10.13.2}$$

where

\begin{align} a_0 \amp = \frac{2C}{L} \int_0^{\frac{L}{2}} dx = C,\tag{10.13.3}\\ a_n \amp = \frac{2C}{L} \int_0^{\frac{L}{2}} \cos\left(\frac{2\pi nx}{L}\right) \,dx = 0 ,\tag{10.13.4}\\ b_n \amp = \frac{2C}{L} \int_0^{\frac{L}{2}} \sin\left(\frac{2\pi nx}{L}\right) \,dx = \begin{cases} \frac{2C}{\pi n} \amp \hbox{($n$ odd)}\\ 0 \amp \hbox{($n$ even)} \end{cases}\tag{10.13.5} \end{align}

Putting this all together,

$$f(x) = C + \sum_{\substack{n=1\\ n\hspace{1.5pt}\mathrm{odd}} }^\infty \frac{2C}{\pi n} \sin\left(\frac{2\pi nx}{L}\right)\text{.}\tag{10.13.6}$$

It is instructive to plot the first few terms of this Fourier series and watch the approximation improve as more terms are included, as shown in Figure 10.13.1.