# Maslov correction harmonic oscillator with imaginary frequency

I have computed the path integral for the harmonic oscillator, in the time sliced formalism. I have been able to reproduce the Maslov correction that appears every half-period. Now I want to compute the path integral for a harmonic oscillator, but now with a frequency $ω^2$ that is purely imaginary. What I get in this case is that there is no Maslov correction, since an imaginary $ω^2$ does not change branch in any of the $N$ Fresnel integral. Is this right?

Sounds right. The Maslov index changes when the classical path has a turning point. If the frequency is pure imaginary, $\omega=i\,w$, $w\in\mathbb{R}$, then the classical EOM is $\ddot{x}=w\,x$. For this EOM, the solution is $x(t)=A\exp(t\surd w)$, which has no turning point.
Edit: now that I think further, one $could$ cook up initial conditions such that you have at most one turning point (e.g. a solution $x(t)=A\exp(-t\surd w)+B\exp(t\surd w)$ where $A>B$). Another way to check your Maslov result, which I think is a lot more straightforward than looking at the branch cuts of the amplitude, is to compute the eigenvalues of the operator formed by the second variation of the Lagrangian.