This, together with , implies that $w_1(0) = 0$, $ w_2(0) = \sqrt{3}$, and $w_3(0) = -\sqrt{3}$. The other relations in Table \[Table:z-w-corres\] can then be verified similarly, and we omit the details here.

This completes the proof of Proposition \[prop:w-expression\].

As a consequence of Proposition \[prop:w-expression\], we have the following properties of $w_j(z)$.

\[prop:w\] The functions $w_j(z)$, $j=1,2,3$, given in satisfy the following properties.

-   $w_j(z)$ is analytic on $\mathcal R_j$, $j=1,2,3$, and $$\begin{aligned}
    w_{1,\pm}(x) &= w_{3,\mp}(x), \qquad x \in (-\infty,-1), \label{eq: w1pm-w3pm}
    \\
    w_{1,\pm}(x) &= w_{2,\mp}(x), \qquad x \in (1,\infty). \label{eq: w1pm-w2pm}\end{aligned}$$ Here, we orient $(-\infty,-1)$ and $(1,\infty)$ from the left to the right. Hence, the function $\cup_{j=1}^3 \mathcal R_j \to {\mathbb{C}}: \mathcal R_j \ni z \mapsto
    w_j(z)$ has an analytic continuation to a meromorphic function $w:
    \mathcal R \to \overline {\mathbb{C}}$. This function is a bijection.

-   $w$ satisfies the symmetry properties $$\begin{aligned}
    w_j(\overline z) &= \overline{w_j(z)},   \qquad~~\,~~\,  z\in\mathcal R_j, \quad j=1,2,3, \label{eq: wz-wzbar} \\
    w_1(-z) & = - w_1(z) ,   \qquad~~\,  z\in  {\mathbb{C}}\setminus \{ (-\infty,-1] \cup [1,+\infty) \} , \label{eq: w1z+-} \\
    w_2(-z)&=-w_3(z),   \qquad~~\,  z \in {\mathbb{C}}\setminus (-\infty,-1]. \label{eq: w2-w3}\end{aligned}$$

-   As $z \to \infty$ and $-\pi<\arg z <\pi$, we have $$\label{eq: w1-large-z}
    w_1(z) =\left\{
              \begin{array}{ll}
                