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The CTFS of $x(t)$ is $c_{k}$ the Fourier series coefficients. Furthermore, $x(t-t_{0})$ is known to be $e^{-j\omega t_{0}}c_{k}$, the proof is given as follow : $$ \begin{aligned} \mathscr{F}\left(f\left(t-t_{0}\right)\right) &=\forall n, n \in \mathbb{Z}:\left(\frac{1}{T} \int_{0}^{T} f\left(t-t_{0}\right) e^{-\left(i \omega_{0} n t\right)} \mathrm{d} t\right) \\ &=\forall n, n \in \mathbb{Z}:\left(\frac{1}{T} \int_{-t_{0}}^{T-t_{0}} f\left(t-t_{0}\right) e^{-\left(i \omega_{0} n\left(t-t_{0}\right)\right)} e^{-\left(i \omega_{0} n t_{0}\right)} \mathrm{d} t\right) \\ &=\forall n, n \in \mathbb{Z}:\left(\frac{1}{T} \int_{-t_{0}}^{T-t_{0}} f(\bar{t}) e^{-\left(i \omega_{0} n \bar{t}\right)} e^{-\left(i \omega_{0} n t_{0}\right)} \mathrm{d} t\right) \\ &=\forall n, n \in \mathbb{Z}:\left(e^{-\left(i \omega_{\mathrm{o}} n \bar{t}\right)} c_{n}\right) \end{aligned} $$ I did not understand what happened in the second step. Furthermore, in the third step why did we replace $t-t_{0}$ by $\overline{t}$ without modifying the boundary of integration and replacing $dt$ by $d\overline{t}$? Any help is much appreciated and thank you.

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Even though Latex is highly encouraged instead of just copying stuff, in this case it would have been interesting to see the original "proof" because it's hard to know which errors come from the original proof and which are just typos introduced by entering the Latex code.

Of course $dt$ should have been replaced, that's just a simple typo. And the integration bounds could have been changed, but that's not necessary in this case because the integrand is $T$-periodic, and hence, as long as you integrate over one period the exact upper and lower bounds don't matter:

$$\int_{0}^Tg(t)dt=\int_{t_0}^{T+t_0}g(t)dt\qquad\text{if}\quad g(t)=g(t+T)$$

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