## 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 $m\text{.}$

###### Activity10.2.1.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.

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.