Error propagation of complex quantity in polar coordinates?

Question

Suppose I have a complex number $C = \left | C \right | e^{i\phi}$ where | | denotes modulus and $\phi$ is the phase angle.

Now, I know the error in the modulus, $\delta |C|$ and error in the phase, $\delta\phi$. What are the error propagation rules to determine the overall error, $\delta C$?

Edit: The "error" I'm referring to is 1 standard deviation.

Answer 1

Error propagation is estimated by differentials, so letting $C = re^{i\phi}$, we have $dC = \frac{\partial C}{\partial r}dr + \frac{\partial C}{\partial \phi}d\phi = \frac {C}{r}dr + iCd\phi$. Or in your notation: $$ \frac{\delta C}{C} = \frac {\delta |C|}{|C|} + i\delta \phi$$