+ \mathcal O(z^{-2} ) \biggr), & {\mathrm{Im}\,}z <0,
   \end{cases}$$

$$p_1(z) = \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}}  \omega^2  z^{-\frac{1}{3}} e^{\theta_2(z;\rho)} \biggl( 1 + \frac{\kappa_3(\rho)}{\omega^2} z^{-\frac{2}{3}} + \frac{\kappa_6(\rho)}{\omega^4} z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr)$$

for $-\frac{3\pi}{4} < \arg z < \frac{5\pi}{4},$ and $$\label{eq:asyp4}
  p_4(z) = \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} z^{-\frac{1}{3}} e^{\theta_3(z;\rho)} \biggl( 1 + \kappa_3(\rho) z^{-\frac{2}{3}} + \kappa_6(\rho) z^{-\frac{4}{3}} + \mathcal O(z^{-2} ) \biggr)$$ for $-\frac{\pi}{4} < \arg z < \frac{7\pi}{4},$ where $\theta_k(z;\rho)$, $k=1,2,3$, are given in , $\kappa_3(\rho)$ and $\kappa_6(\rho)$ are polynomials in $\rho$ given in and . The asymptotic expansions of $p_j'(z)$ and $p_j''(z)$, $j=0,1,4$, can be derived in a similar fashion or simply by taking derivatives on the right hand sides of –. A combination of all these asymptotic results and then gives us after a straightforward calculation.

Now, define $$\label{eq: tilde-psi}
\widetilde \Psi (z)=\widetilde \Psi (z;\rho)=
\begin{pmatrix}
p_0(z) & p_1(z) & p_4(z)
\\
p_0'(z) & p_1'(z) & p_4'(z)
\\
p_0''(z) & p_1''(z) & p_4''(z)
\end{pmatrix}, \qquad z\in \mathbb{C},$$ that is, $\widetilde \Psi$ is the analytic extension of the restriction of $\Psi$ on the region $\Theta_1$ to the whole complex plane. The Pearcey kernel then admits the following equivalent representation in terms of $\widetilde \Psi$ (see [@BK3 Equation (10.19)]): $$\label{eq:PearceyRH}
K^{\mathrm{Pe}}(x,y;\rho)=\frac{1}{2 \pi i(x-y)}
\begin{pmatrix}
