node \[above\] (q1) [$-1$]{}; (5,0.5) circle (2pt) node \[above\] (q1) [$1$]{}; (5,-1.5) circle (2pt) node \[above\] (q1) ; (3,-3.5) circle (2pt) node \[above\] (q1) ;

We intend to find functions $\lambda_j$, $j=1,2,3$, on these sheets, such that each $\lambda_j$ is analytic on $\mathcal R_j$ and admits an analytic continuation across the cuts. For this purpose, we start with an elementary function $w(z)$ that is meromorphic on $\mathcal R$, which satisfies the following algebraic equation $$\label{eq: alg eq w}
w(z)^3-3w(z)+2z=0.$$ We choose three solutions $w_j(z)$, $j = 1,2,3$, to such that they are defined and analytic on $\mathcal R_j$, respectively. Each function $w_j(z)$ maps $\mathcal R_j$ to certain domain in the extended complex $w$-plane $\overline {\mathbb{C}}$. The correspondences between some points $z\in \mathcal R$ and the points $w \in \overline {\mathbb{C}}$ are given in Table \[Table:z-w-corres\], where $z^{(j)}$ denotes the point $z$ on the closure of the $j$-th sheet $\mathcal R_j$.

  ----------------------------------------- ------------ ---------------- ----------- ----------- ------------ -------------
                                                                                                               
         \[-1em\] $z \in \mathcal R$        $-1^{(1)}$   $\infty^{(1)}$   $1^{(1)}$   $0^{(1)}$   $0^{(2)}$    $0^{(3)}$
                   \[1ex\]                                                                                     
   \[-1em\] $w \in \overline {\mathbb{C}}$  $-1$         $\infty$         1           0           $\sqrt{3}$   $-\sqrt{3}$
                   \[1ex\]                                                                                     
  ----------------------------------------- ------------ ---------------- ----------- ----------- ------------ -------------

  : Images of some points $z\in \mathcal R$ under the mappings $w_j$, 