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Section 10.1 Fourier Series Motivation

If a room full of students is asked to sketch an example of a periodic function, at least half will draw a sine or cosine. This is a great choice in the sense that sines and cosines are particularly simple and easy to deal with algebraically. (Even easier are complex exponentials – more about this later.) However, there are many other periodic functions that are not so simple. These functions can represent oscillatory phenomena that occur in many places in nature. Fortunately for us, it turns out that we can express all periodic phenomena as (possibly infinite) sums of sines and cosines. These sums are called Fourier Series.

Here is the basic formula for a Fourier series for the periodic function \(f(\theta)\) with period \(2\pi\text{,}\) that is, if \(f(\theta+2\pi)=f(\theta)\text{:}\)

\begin{equation} f(\theta) = \frac12 a_0 + \sum_{n=1}^\infty a_n \cos n\theta + \sum_{n=1}^\infty b_n \sin n\theta\tag{10.1.1} \end{equation}

The extra factor of \(\frac{1}{2}\) in the \(a_0\) term is a convention. Also, there is no \(b_0\) term since \(\sin{0}\) is identically zero.

The coefficients \(a_n\) and \(b_n\) are uniquely determined by \(f(\theta)\text{.}\) The formulas for finding the coefficients for a given \(f(\theta)\) can be found in Section 10.5. The geometric idea behind the formulas for the coefficients can be found in Section 8.5.

Fourier Series are often used to approximate a function by giving the first few terms. In Figure 10.1 you can see how well the first few terms (shown in purple) approximate a periodic step function (shown in blue), with period \(L=1\text{.}\) You can also use this applet to approximate another function of your choice by entering it in the box labeled “Function”. (The applet will automatically convert your function into a periodic function with period \(1\text{.}\))

Figure 10.1. The first few partial sums in the Fourier series for a step function. Move the slider to increase the number of terms in the sum.