# Unable to understand the time-shifting property of CTFS

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.

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