First off, I'm not sure if this question belongs here so I apologize if it's out of place.
I'm working with a system of coupled linear partial differential equations in space and time with which I'd like to conduct frequency response analysis. I took a LaPlace transform of the system with respect to time, solved a resulting second order ODE for x, and ended up with a complicated transfer function in $x$ and $s$. It looks like this:
$F\left(s\right)=\frac{\text{Stuff}}{e^{s+\sqrt{As^2+Bs+c}}+...}$
Where the exponential term has that form because it is one of the roots of the characteristic equation for the ODE in $x$.
Frequency response analysis techniques then say to replace instances of $s$ with $j\omega$, where $\omega$ is the frequency of the input sine wave, and then plot the amplitude and angle of $F\left(j\omega\right)$. My question then is is there an "informed" choice to make for which square root solution I should take when computing these values.
The analog that comes to mind for me is when we take roots with real numbers, like when we calculated the roots of the characteristic equation of the above described ODE, we take the positive root and then write plus/minus to indicate roots. Here though since we're rooting a complex number we can have 4 roots and I'm not sure which one to take. When we aren't working in the complex plane, we'll often only consider the positive root because it has physical meaning. I'm just not sure if there is a similar interpretation here.