Fourier Series: Shifting in time domain I am reading "Fourier Transformation for Pedestrians" from T. Butz. He speaks about what happens to the Fourier coefficients when the function is shift in time. I have copied the equation I have a problem with:

I don't understand the logic behind going from $f(t-a)$ in the first integral to $f(t')e^{-iw_kt'}e^{-iwk_a}dt'$. It seems like he is using the identity $e^{a+b}=e^ae^b$ but I don't understand the complete logic.
Also he then applies the same thing without the complex notation:
 
Why does that work? Why does shifting in time correspond to multiplying $A_k$ for example by $cos \omega_k a - B_k sin \omega_k a$. I don't understand where this is the case. If someone could explain it would be great. Thank you.

In the first step you have a question about, he replaces $t-a$ with $t'$. So, everywhere $t$ appears it becomes $t' + a$. Thus


*

*$f(t -a)$ becomes $f(t')$

*$e^{-i\omega_k t}$ becomes $e^{-i\omega_k (t'+1)} = e^{-i\omega_k t'} e^{-i\omega_k a}$

