I am having trouble solving this questions:
"Let $\sum\limits_{n=-\infty}^\infty{{c_n}{e^{-inx}}} $ be the Fourier series of $f(x)$. find the Fourier series of $g(x)=f(x+a)$".
I have tried to start from the definition for $g(x)$ complex coefficients ($D_n$) and extract $f(x)$ complex coefficients ($C_n$).
I have ended up with an expression very close to what I want but with non matching integration limits. I would appreciate any suggestions or solution to this problem :) .
what I have so far:
$D_n$ $= \frac{1}{{2\pi}} \int\limits_{-\pi}^\pi{g(x)e^{-inx}}dx $ $= \frac{1}{{2\pi}} \int\limits_{-\pi}^\pi{f(x+a)e^{-inx}}dx$ $= \begin{Bmatrix}t=x+a\\ds=dt\\ \end{Bmatrix}$ $= \frac{1}{{2\pi}} \int\limits_{-\pi+a}^{\pi+a}{f(t)e^{-in(t-a)}}dt$ $= \frac{e^{ina}}{{2\pi}} \int\limits_{-\pi+a}^{\pi+a}{f(t)e^{-int}}dt$
The integral is an interval that has length $2\pi$ so it gives you the wanted result at the end $e^{ina}C_n$.