Phase Shift in Fourier Series

Question

So I came across a problem that has a square wave shifted in time corresponding to a $\frac{\pi}{4}$. We constructed the Fourier Series using the formulation $f(x)=\sum{C_n}e^{inwt}$, where we are summing from $n=-\infty$ to $\infty$, $w$ is the fundamental frequency of the signal, and $C_n$ is complex and contains the phase information. Now I was taught that to find the new $C_n$ corresponding to the phase shift, I would simply multiply the old $C_n$ by a phase factor $e^{i\phi}$ and I will get the desired answer. Is this always true? Is there a way to prove this? Thanks!

Answer 1

Suppose the wave was shifted by a phase $\theta$. Then, if we shift our time coordinates from $t$ to $t-\frac{\theta}{\omega}$, we can transform the original wave to a new wave.

Thus, if we let $t' = t-\frac{\theta}{\omega}$ as the time coordinate for the shifted wave, then we have - $f(t') = \sum{C_n}e^{in(\omega t'+\theta)} = \sum{C_ne^{in\theta}}e^{in\omega t'}$.

So, the new Fourier coefficients are just a product of the original ones and a phase shift.

(Note that this holds for any wave, and not just a square one.)