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

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*}

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 $$f(x)=\frac{x-2k\pi}{\pi}$$.

• Ok, that clears a lot up, but then what is the function who's fourier series I need to find? Is it the last f(x) you put? Can I still find its fourier series if it has 'x' and 'k' in it? Apr 19, 2021 at 13:22
• It is the linear function equal to $-1$ at $-\pi$, and $1$ at $\pi$, reproduced periodically outside the interval $(-\pi,\pi)$ with the global formula $\;f(x)=f(x\bmod 2\pi)$. Apr 19, 2021 at 14:08
• @Maximus I will say it again: THERE IS ONLY ONE FUNCTION.
– 5xum
Apr 20, 2021 at 6:18