I'm wondering if its correct (and possible to prove) that the phase of the mean of a set of complex numbers is equivalent to the weighted average of the phases of the set where the weights are the magnitudes.
Given a set of complex numbers $z_j$ we have \begin{align} z_j &= x_j + iy_j = r_j e^{i\theta_j}\\ \langle z \rangle &= \langle x \rangle + i \langle y \rangle = \bar{r} e^{i\bar{\theta}} \end{align}
Where the expected value of a variable ($\langle x \rangle$) has its usual meaning and $\bar{r}$ and $\bar{\theta}$ have the following definitions:
\begin{align} \bar{r} &= \sqrt{\langle x \rangle^2 + \langle y \rangle^2}\\ \bar{\theta} &= \arctan\frac{\langle y \rangle}{\langle x \rangle} \end{align}
My question at the beginning then becomes:
\begin{equation} \bar{\theta} \stackrel{?}{=} \frac{\langle r\theta \rangle}{\langle r\rangle} \end{equation}
I've tried expanding either side of the equation but an arctan always shows up that I can't get rid of, any ideas?