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$$

f(\theta) =

\begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\

-1 & \text{if} \ -\pi < \theta < 0

\end{cases}.

$$

$f$ is periodic with period $2\pi$ and odd since $f$ is symmetric about the origin.

So $f(-\theta) = -f(\theta)$.

Let $f(\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{in\theta}$.

Then $f(-\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{-in\theta} = \cdots + a_{-2}e^{2i\theta} + a_{-1}e^{i\theta} + a_0 + a_{1}e^{-i\theta} + a_{2}e^{-2i\theta}+\cdots$

$-f(\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{-in\theta} = \cdots - a_{-2}e^{-2i\theta} - a_{-1}e^{-i\theta} - a_0 - a_{1}e^{i\theta} - a_{2}e^{2i\theta}-\cdots$

$a_0 = -a_0 = 0$

I have solved many Fourier coefficients but I can't think today.

What do I need to do next?