 Solution

For $\theta >1$, the stationary-point equation would requireso the two saddles leave the real axis. The contour can be deformed through the appropriate complex saddle or saddles, and their contributions become exponentially small or large according to the sign of the real part of $i\Phi$. The real-axis oscillatory saddle contributions present for $0<\theta <1$ therefore disappear across the turning point $\theta =1$. The admissible deformation and the pole's residue must still respect the prescribed branch cuts and the instruction that the contour pass above $z_0$.