Uncertainty Propagation of Complex Functions

[HermitianField]

Suppose I have some observables $\alpha, \beta, \gamma$ whose central values and uncertainties $\sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma}$ are known.

Define a function $f(\alpha, \beta, \gamma)$ which has both real and complex parts. How do I do standard error propagation when imaginary numbers are involved? This problem deals with the eigenvalues of a coupled oscillator. Here, some of the eigenvalue functions are complex, so I would like to know how to calculate the uncertainty on f, which is an eigenvalue. The claim is that since coupled oscillators are physical systems, their answers are real in nature. Thus, should I use $Re(f)$ instead?

 

[Dale]

Some physical systems are better represented by complex numbers than real numbers. So simply being a physical system doesn't imply the measurement is a real number.

The better question is what is the distribution of the complex number and the function.