and $$\begin{aligned}
\frac{\partial}{\partial s} F(s;\rho)&=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}s} \ln \det\left(I-K_{s,\rho}^{{\mathrm{Pe}}}\right) \nonumber
\\
&=-\frac{1}{ \pi i} \left[\lim_{z \to -s} \left(\left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}\right)
\right],
\label{eq:derivativeinsX-2}\end{aligned}$$ where the above limit is taken from the region $\texttt{II} \, \cup \texttt{V}$.

We begin with the proof of . From and , it follows that $$\mathsf{Y}_1 = \int_{-s}^s {\mathbf}{F}(v){\mathbf}{h}^t(v) {\,\mathrm{d}}v  = \int_{-s}^s \begin{pmatrix}
F_1(v) \\
F_2(v) \\
F_3(v)
\end{pmatrix} \begin{pmatrix}
h_1(v) & h_2(v) & h_3(v)
\end{pmatrix} {\,\mathrm{d}}v .$$ This, together with , implies that $$\label{eq:Frho1}
    \frac{\partial}{\partial \rho} F(s;\rho) = - \frac{1}{2} \left[ (\mathsf{Y}_1)_{12} + (\mathsf{Y}_1)_{23} \right].$$ By , it is readily seen that $\mathsf{Y}_1 = \Psi_0 (\mathsf{X}_1 - \Psi_1 ) \Psi_0^{-1}$. With the aid of the explicit expressions of $\Psi_0$ and $\Psi_1$ in , we then have $(\mathsf{Y}_1)_{12} + (\mathsf{Y}_1)_{23}=(\mathsf{X}_1)_{12} + (\mathsf{X}_1)_{23}-\frac{\rho^3}{27}$, which gives us in view of .

We next consider $\frac{\partial}{\partial s} F(s;\rho)$. For $z\in \texttt{II}$, we see from , and that $$\label{eq:FHinX}
{\mathbf}{F}(z)=Y(z){\mathbf}{f}(z)=Y(z)\widetilde \Psi(z)\begin{pmatrix}
1
\\
0
\\
0
\end{pmatrix}=X(z)
\begin{pmatrix}
1
\\
0
\\
0
\end{pmatrix}$$ and $${\mathbf}{H}(z)=Y^{-t}(z){\mathbf}{h}(z)=X^{-t}(z)\widetilde \Psi^{t}(z)\cdot \frac{\widetilde \Psi^{-t}(z)}{2 \pi i}\begin{pmatrix}
0
\\
1
\\
1
\end{pmatrix}=\frac{X^{-t}(z)}{2 \pi i}
\begin{pmatrix}
0
\\
1
\\
1
\end{pmatrix}.$$ Combining the above formulas, , and L’Hôspital’s rule then gives us $$\begin{aligned}
\frac{\partial}{\partial s} F(s;\rho)
=&-\frac{1}{2 \pi i} \bigg[\lim_{z 