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Section 10.2 Fourier Basis Functions

Before we jump into the details of Fourier series, use the applet below to remind yourself of how the graphs of \(\sin\left(\frac{2\pi m x}{L}\right)\) and \(\cos\left(\frac{2\pi m x}{L}\right)\) depend on the parameter \(m\text{.}\)

Activity 10.2.1. Visualizing the Fourier Basis Functions.

Use the applet below to explore the basis functions. One at a time, set each slider to \(1\text{,}\) look at the resulting function, and return that slider to \(0\text{.}\) Make a note of any patterns that you see.

Figure 10.2.1. An applet for manipulating the individual Fourier coefficients (with \(L=1\)).

You should have noticed the following features:

  1. All of the functions are sinusoidal, that is, \(\cos(m\pi x)\) or \(\sin(m\pi x)\) for integer \(m\text{.}\)
  2. The coefficient of the constant term (\(\cos(m\pi x)\) for \(m=0\)) is \(\frac12 a_0\text{.}\)
  3. The coefficients \(a_m\) correspond to cosine functions.
  4. The coefficients \(b_m\) correspond to sine functions.
  5. The subscript \(m\) on the coefficients \(a_m\) and \(b_m\) corresponds to the value \(m\) in \(\cos(m\pi x)\) or \(\sin(m\pi x)\text{.}\)

The basic idea of Fourier series is that any periodic function with period \(L\) can be built up as a linear combination of the functions \(\cos\left(\frac{2\pi m x}{L}\right)\) or \(\sin\left(\frac{2\pi m x}{L}\right)\text{.}\) For any given function, the linear combination is unique.