$j=1,2,3$.[]{data-label="Table:z-w-corres"}

Due to the sheet structure shown in Figure \[fig: Riemann surface\], it follows that $\infty^{(1)} = \infty^{(2)} = \infty^{(3)}$, $-1^{(1)} = -1^{(3)}$ and $1^{(1)} = 1^{(2)}$. Moreover, we actually have the following explicit expressions for $w_j(z)$.

\[prop:w-expression\] The three solutions $w_j(z)$, $j=1,2,3$, to the algebraic equation with the mapping properties shown in Table \[Table:z-w-corres\] are given by $$\label{eq: w-eta}
  w_j(z) = \omega^{j-2}\eta (z)^{\frac{1}{3}} + \omega^{2-j}\eta (z)^{-\frac{1}{3}},$$ where $$\label{eq: eta-def}
  \eta(z) = (z^2 -1 )^{\frac{1}{2}} - z, \qquad z \in \mathbb{C} \setminus \{ (-\infty, -1] \cup [1,+\infty)\},$$ with $$\label{eq: eta-arg}
  \arg \eta(z) \in (0, \pi).$$

Clearly, $\eta(z)\neq 0$ and satisfies the quadratic equation $$\label{eq:etaeq}
    \eta(z)^2 + 2z \eta(z) + 1 = 0.$$ Hence, with $w_j(z)$, $j=1,2,3$, defined in , we have $$\begin{aligned}
    w_j(z)^3 -3w_j(z) + 2z & =  \left( \omega^{j-2}\eta (z)^{\frac{1}{3}} + \omega^{2-j}\eta (z)^{-\frac{1}{3}} \right)^3 - 3 \left( \omega^{j-2}\eta (z)^{\frac{1}{3}} + \omega^{2-j}\eta (z)^{-\frac{1}{3}} \right) + 2z
     \\
     & =  \eta(z) + \frac{1}{\eta(z)} + 2z = 0,
  \end{aligned}$$ as expected, where in the last step we have made use of . Furthermore, an elementary analysis shows that ${\mathrm{Im}\,}\eta(z) >0$ for $z \in \mathbb{C} \setminus \{ (-\infty, -1] \cup [1,+\infty)\}$. Thus, $\arg \eta(z) \in (0, \pi)$ and $\eta(0) = e^{\frac{\pi i}{2}}=i $. 