Fourier Series of a piecewise function

Question

I have to find the Fourier series of the following function:

$$ f(x)= \begin{cases} 1 & \pi/2 < |x| < \pi \\ 0 & otherwise \\ \end{cases} $$ I don't understand how to find the bounds for the integration (to find $b_k$ and $a_k$).

Could somebody please help me? Thank you!

Answer 1

Let $m\in \Bbb{Z}$

$\pi \geq|x| \geq \pi/2$ if and only if $x \in [-\pi,-\pi/2]\cup [\pi/2,\pi]$

Also note that $e^{ix}=\cos{x}+i\sin{x}$

Then $$\hat{f}(m)=\frac{1}{2\pi}\int_{-\pi}^{-\frac{\pi}{2}}e^{-imx}dx+\frac{1}{2\pi}\int_{-\frac{\pi}{2}}^{\pi}e^{-imx}dx$$

So from this you can find $a_m,b_m$