& p_4''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_2$,} \\
          \begin{pmatrix} p_4(z) & -p_5(z) & p_3(z)\\ p_4'(z) & -p_5'(z) & p_3'(z) \\ p_4''(z) & -p_5''(z) & p_3''(z)\end{pmatrix}, & \quad \hbox{$z\in \Theta_3$,} \\
          \begin{pmatrix} p_0(z) & p_2(z) & p_3(z)\\ p_0'(z) & p_2'(z) & p_3'(z) \\ p_0''(z) & p_2''(z) & p_3''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_4$,} \\
          \begin{pmatrix} p_1(z) & p_2(z) & p_5(z)\\ p_1'(z) & p_2'(z) & p_5'(z) \\ p_1''(z) & p_2''(z) & p_5''(z) \end{pmatrix}, & \quad \hbox{$z\in \Theta_5$,}
        \end{array}
      \right.$$ where $\Theta_k$, $k=0,1,\ldots,5$, is the region bounded by the rays $\Sigma_k$ and $\Sigma_{k+1}$ (with $\Sigma_6:=\Sigma_0$); see Figure \[fig: sigma\] for an illustration.

For our purpose, we present a refined asymptotics of $\Psi$ at infinity in , which can be verified directly from . To see this, let us focus on the case $z\in\Theta_1$, since the asymptotics in other regions can be derived in a similar way. Carrying out a steepest descent analysis to the integrals defined (cf. [@BK3; @Miyamoto]), we have, as $z\to \infty$,

$$\label{eq:asyp0}
  p_0(z) = \begin{cases}
     -\sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}}  \omega  z^{-\frac{1}{3}} e^{\theta_1(z;\rho)} \biggl( 1 + \frac{\kappa_3(\rho)}{\omega} z^{-\frac{2}{3}} + \frac{\kappa_6(\rho)}{\omega^2} z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr), & {\mathrm{Im}\,}z >0, \\
      \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}}  \omega^2  z^{-\frac{1}{3}} e^{\theta_2(z;\rho)} \biggl( 1 + \frac{\kappa_3(\rho)}{\omega^2} z^{-\frac{2}{3}} + \frac{\kappa_6(\rho)}{\omega^4} z^{-\frac{4}{3}} 