Fourier Series Motivation

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Section 17.1 Fourier Series Motivation

UNDER DEVELOPMENT--Add the example of the Fourier series for a pendulum that is NOT in the small angle approximation. Change the figure below.

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.

Fourier Series are often used to approximate a function by giving the first few terms. In Figure 17.1 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 17.1. 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.