$X(z)$ given in , we further obtain from and a bit more cumbersome calculation that $$X(z)\widetilde X^{-1}(z) = \widetilde{C} + {\mathcal{O}}(z^{-1}), \qquad \textrm{as } z \to \infty,$$ with $\widetilde{C}$ given in . An appeal to Liouville’s theorem then leads us to .

As a consequence of , we have $$X'(z) = -\widetilde {C} X'(-z) {\Lambda}.$$ This, together with and the fact that ${\Lambda}^{-1}={\Lambda}$, implies $$\begin{aligned}
  \lim_{z \to s} \left( X^{-1}(z)X'(z) \right) =  - \lim_{z \to s} \left( {\Lambda}X^{-1}(-z) X'(-z) {\Lambda}\right)  =  - \lim_{z \to -s} \left( {\Lambda}X^{-1}(z) X'(z) {\Lambda}\right) .\end{aligned}$$ To this end, we observe that for an arbitrary $3\times 3$ matrix $ M = (m_{ij})_{i,j=1}^3$, $$\left( {\Lambda}M {\Lambda}\right)_{21} = - m_{31} \quad \textrm{and} \quad  \left( {\Lambda}M {\Lambda}\right)_{31} = - m_{21}.$$ A combination of the above two formulas shows that $$\lim_{z \to -s} \left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}
=
\lim_{z \to s} \left(X^{-1}(z)X'(z)\right)_{21}+\left(X^{-1}(z)X'(z)\right)_{31}.$$ Inserting this formula into gives us .

This completes the proof of Proposition \[prop:derivativeandX\].

Auxiliary functions {#sec:auxifuncs}
===================

\[sec:auxiliary function\]

In this section, we introduce some auxiliary functions and study their properties. The aim is to construct the so-called $\lambda$-functions, of which the analytic continuation defines a meromorphic function on a Riemann surface with a specified sheet structure. The $\lambda$-functions have 