How to write this functions as a Fourier series

Question

Let $f :[0,2\pi] \rightarrow \mathbb{R}$ of $C^1$ and $2\pi-$periodic

if $f$ is definied as :

$f(\theta)=f_1(\theta)\quad \quad$ for $ \theta \in [0,\frac{\pi}{3}]$

$f(\theta)=f_2(\theta)\quad\quad$ for $\theta \in [\frac{\pi}{3},\frac{2\pi}{3}]$

$f(\theta)=f_3(\theta) \quad\quad$ for $\theta \in [\frac{2\pi}{3},2\pi]$

and I want to write $f$ as a fourier series $\sum_{n} a_n e^{ i n\theta}$ ,

I have to write each composante $f_1,f_2$and $f_3$ as a fourirer series or there another way to do it ?

thank you in advance

Answer 1

$$a_n = \frac{1}{2\pi} \int_0^{2\pi} e^{-int}f(t)$$

$$= \frac{1}{2\pi} \left(\int_0^{\frac{\pi}{3}} e^{-int}f_1(t) + \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} e^{-int}f_2(t) + \int_{\frac{2\pi}{3}}^{2\pi} e^{-int}f_3(t) \right )$$