Calculate Fourier series of $f$, but there are multiple $f(x)$'s.

Question

I've been asked to calculate the Fourier series for $f$, however, I've been given two $f(x)$'s (see below). I know how to calculate Fourier coefficients and the Fourier series but in this case, I'm not sure what to do since there are multiple $f(x)$'s. Do I just calculate the first one? Or do I need to do both? \begin{align*} & f(x) = \frac{x}{\pi },\text{ if } - \pi \le x < \pi , \\ & f(x) = f(x + 2\pi ). \end{align*}

Answer 1

That's not multiple $f(x)$s, there is just one. $f$ is a periodic function with a period of $2\pi$.

In particular:

• on the interval $[-\pi, \pi)$, the value of $f(x)$ is $\frac{x}{\pi}$.
• On the interval $[-3\pi, -\pi)$, the value of $f(x)$ is equal to $f(x)=f(x+2\pi)=\frac{x+2\pi}{\pi}$.
• On the interval $[\pi, 3\pi)$, the value of $f(x)$ is (because you can show that $f(x)=f(x-2\pi))$ $f(x)=\frac{x-2\pi}{\pi}$.

In general, on the interval $[(2k-1)\pi, (2k+1)\pi)$, the value of $f(x)$ is equal to $f(x)=\frac{x-2k\pi}{\pi}$.