What is the Fourier series of $f(x) = \begin{cases} -2, & -4<x<-2 \\ x, & {-2<x<2} \\ 2, & {2<x<4} \end{cases}$

Question

What is the Fourier series of $f(x) = \begin{cases} -2, & -4<x<-2 \\ x, & {-2<x<2} \\ 2, & {2<x<4} \end{cases}$

I have already established that this function is an odd function and has a period of 8, however, I do not know how to move forward with this. Any explanation on what definition of $f(x)$ should I follow would be greatly appreciated.

Answer 1

Hint.

If the function $f$ is odd on the symmetric interval $[-L,L]$, then you have the Fourier sine series

$$ f(x)\sim \sum_{n=1}^\infty a_n\sin (\frac{n\pi x}{L}) $$ where the coefficients are given by $$ a_n=\frac{2}{L}\int_0^L f(x)\sin (\frac{n\pi x}{L})dx $$

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Notes.

In your example,

$$ \int_0^4 f(x)\sin (\frac{n\pi x}{4})dx =\int_0^2x\sin (\frac{n\pi x}{4})dx +\int_2^42\sin (\frac{n\pi x}{4})dx $$