## Section10.2Fourier 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 $$m\text{.}$$

### Activity10.1.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.

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.