desired behavior around each branch point, and will be crucial in our further asymptotic analysis of the RH problem \[rhp:X\] for $X$.

Throughout this section, unless specified differently, we shall take the principal branch for all fractional powers.

A three-sheeted Riemann surface and the $w$-functions {#sec:w function}
-----------------------------------------------------

We introduce a three-sheeted Riemann surface $\mathcal R$ with sheets $$\begin{aligned}
\mathcal R_1 &= {\mathbb{C}}\setminus \{(-\infty,-1]\cup[1,+\infty) \}, & \mathcal R_2 &= {\mathbb{C}}\setminus [1,+\infty), \\
\mathcal R_3 &={\mathbb{C}}\setminus (-\infty,-1] .\end{aligned}$$ We connect the sheets $\mathcal R_j$, $j=1,2,3$, to each other in the usual crosswise manner along the cuts $(-\infty,-1]$ and $[1,+\infty)$. More precisely, $\mathcal R_1$ is connected to $\mathcal R_2$ along the cut $[1,+\infty)$ and $\mathcal R_1$ is connected to $\mathcal R_3$ along the cut $(-\infty,-1]$. We then compactify the resulting surface by adding a common point at $\infty$ to the sheets $\mathcal R_1$ and $\mathcal R_2$, and a common point at $\infty$ to the sheets $\mathcal R_1$ and $\mathcal R_3$. We denote this compact Riemann surface by $\mathcal R$, which has genus zero and is shown in Figure \[fig: Riemann surface\].

(0,0)–(6,0)–(8,1)–(2,1)–cycle (0,-2)–(6,-2)–(8,-1)–(2,-1)–cycle (0,-4)–(6,-4)–(8,-3)–(2,-3)–cycle; (1,0.5)–(3,0.5) (5,0.5)–(7,0.5) (1,-3.5)–(3,-3.5) (5,-1.5)–(7,-1.5); (8,0.5) node[$\mathcal R_1$]{} (8,-1.5) node[$\mathcal R_2$]{} (8,-3.5) node[$\mathcal R_3$]{}; (3,0.5)–(3,-3.5); (5,0.5)–(5,-1.5); (3,0.5) circle (2pt) 