I'm quite struggling with the following problem:
-Prove that if \begin{equation} re^{i\psi} = \frac{1}{N}\sum_{j=1}^N e^{i\theta_{j}}, \end{equation}
then $\psi = \frac{1}{N}\sum_{j=1}^N \theta_{j}$, i.e the phase of the average of unitary complex numbers is the average phase
I managed to prove it for N=2, but I need a proof for arbitrary N.
After a bit of rumination, I've come up with a possible answer.
If the phases $\theta_j$ are clustered near each other, we could take the paraxial approximations
\begin{equation} \psi - \theta_i = arctg\left[\frac{\sum_{j=1}^Nsin(\theta_j-\theta_i)}{\sum_{j=1}^Ncos(\theta_j-\theta_i)}\right] \approx arctg\left[\frac{\sum_{j=1}^N(\theta_j-\theta_i)}{N}\right] \approx \frac{\sum_{j=1}^N(\theta_j-\theta_i)}{N}, \end{equation} so that \begin{equation} \psi \approx \frac{1}{N}\sum_{j=1}^N\theta_j, \end{equation} which could be the reason why $\psi$ is usually taken as a measure for the average phase. If that is the case, this estimate should be considered only for quasi-coherent states.