\to -s} \left(\left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}\right)
\nonumber
\\
& +
\lim_{z \to s} \left(\left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}\right)\bigg],
\label{eq:derivativeinsX}
$$ where the above limits are taken from the region $\texttt{II}$. Similarly, one can show that also holds provided the limits are taken from the region $\texttt{V}$.

The expression can be further simplified via the following symmetric relation of $X(z)$: $$\label{eq: Xz+-}
  X(z) = \widetilde {C} X(-z) {\Lambda},$$ where $$\label{def:tildeC}
  \widetilde {C}= \Psi_0 \begin{pmatrix}
    -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1
  \end{pmatrix} \Psi_0^{-1}$$ is a nonsingular matrix with $\Psi_0$ given in , and $$\label{eq: MatLam}
     {\Lambda}=
\begin{pmatrix}
-1 & 0 & 0
\\
0 & 0 & 1
\\
0 & 1 & 0
\end{pmatrix}.$$ To see , let us consider the function $$\label{def:tildeX}
  \widetilde X(z):=X(-z) {\Lambda}.$$ It is straightforward to check that $\widetilde X(z)$ satisfies the same jump condition as $X(z)$ shown in and . This means $X(z)\widetilde X^{-1}(z)$ is analytic in the complex plane with possible isolated singular points located at $z = \pm s$. Since $ X(z) = \mathcal O(\ln(z \mp s)) $ as $z \to \pm s$, we conclude that the possible singular points $\pm s$ are removable. In view of the asymptotics of 