In attached figure, $\psi_{\mp}(z)\sim e^{\mp az^4}$, where $z \in \mathbb{C} $and $a> 0$. These two function go to zero at $\infty$ in regions( shaded and unshaded Stokes sectors) as shown in that figure.
Now let $a=\rho e^{i \theta}$, and allow rotation from $\theta=0$ to $\theta=\pi$, but centre of shaded region, i.e., line $~Im(z)=0$, gets rotated Anticlockwise by $\theta=\pi/4$, to the centre of Unshaded region, i.e., $Re(z)=-Im(z)$, why not in clock-wise direction ??
If I am able to express problem, then can anyone help me in that??
Basically it is about analytical continuation of $\psi_{-}(z)\sim e^{- az^4}$, from $a$ to $-a$
One can see the Eq. 11 to 14. of this paper for better understanding of Ques