I'm out of my league. Apologies in advance for the poorly formulated question; I'd be happy to take advice on how to express myself more clearly.
I'm having a very difficult time trying to select the correct root when working with complex numbers. In the work I am doing, there are two different ways that people express "the same" information:
1) The "$\eta$-format": $\eta = n + i\kappa$
2) The "$\epsilon$-format", which is the square of the "$\eta$-format": $\epsilon = \eta^2 = (n^2 - \kappa^2) + i(2n\kappa)$
Often, I want to convert from the $\epsilon$-format back to the $\eta$-format. This presents a problem, since the sign information of the original number has not been preserved through the squaring operation. If I take the principal square root of the square, I will always end up with a positive real part, by definition. If I take the secondary square root of the square, I always end up with a negative real part, by definition. Is there a method to ensure that you end up choosing the "correct" (original) root when square-rooting a complex number? You can assume that I have access to the original $\eta$-format complex number.
The problem is not simply going back and forth between squaring and square-rooting (a rather pointless exercise), but rather the following:
a) taking several $\eta$-format inputs and converting them to $\epsilon$-format
b) "doing some math" on the $\epsilon$-format numbers (effective medium approximation, but that's probably off topic). The equations are formatted in terms of $\epsilon$-format numbers, and I doubt I'm clever enough to convert them to be formatted in terms of $\eta$-format numbers.
c) trying to convert back to $\eta$-format
I don't know which root to select, because I am combining several $\eta$-format complex numbers, some of which may have positive real parts, and others of which may have negative real parts. Thanks for your consideration.