I'm trying to find the Fourier series expansion for $f(\theta)=\cos(\theta/2)$ on $[-\pi,\pi]$. To solve for the coefficient, I'm trying
$$ c_k=\int_{-\pi}^\pi\cos\left(\frac{\theta}{2}\right)e^{-ik\theta}\frac{d\theta}{2\pi} $$ I got stuck on solving this integral. Any suggestions on how I can do that? Thanks!
You can work with trig functions as in the hint, or with complex exponentials.
Or try both and check that the results agree!
$$\int_{-\pi}^\pi \cos \left( \frac{\theta}{2} \right) e^{-ik\theta} d\theta = \frac{1}{2}\int_{-\pi}^\pi \left[e^{i\theta/2}+e^{-i\theta/2}\right] e^{-ik\theta} d\theta =\frac{1}{2}\int_{-\pi}^\pi \left[e^{i(1/2-k)\theta}+e^{-i(1/2+k)\theta}\right] d\theta $$