be discontinuous across the branch cuts joining these branch points. Moreover, from (\[eq:zeroasym\]), one of the functions $\xi_j(z)$ will have a simple pole at $z=0$. Apart from these singularities, however, the functions $\xi_j(z)$ are analytic.

Sheet structure of the Riemann surface
--------------------------------------

As explained in the introduction, the determination of the sheet structure of the Riemann surface (\[eq:curve20\]) involves difficult analysis of the behavior of the $\xi_j(z)$ at the points $\lambda_k$. In order to implement Riemann-Hilbert analysis to obtain the asymptotics of the kernel in (\[eq:mpoint\]), we must show that $\xi_1(z)$ is not analytic at the real branch points $\lambda_k$, $k=1,2$. This will be achieved by making use of the properties of the Stieltjes transform.

In this section we will study the sheet structure of the Riemann surface ${\mathcal{L}}$. As we shall see, the branch $\xi_1(z)$ turns out to be the Stieljes transform $m_{\underline{F}}(z)$ and its branch cut will be the support of $\underline{F}$.

By Lemma \[le:CS\], the real points at which $\frac{d z}{d\xi}>0$ characterize the end points of $\mathrm{Supp}(\underline{F})$. The determine the support, let us differentiate (\[eq:curve1\]) to obtain an expression of $\frac{d z}{d\xi}$ in terms of $\xi$. $$\label{eq:zprime}
\begin{split}
\frac{d
z}{d\xi}&=\frac{1}{\xi^2(1+\xi)^2(1+a\xi)^2}\Bigg(a^2(1-c)\xi^4
+2(a^2(1-c\beta)+a(1-c(1-\beta))\xi^3\\
&+(1-c(1-\beta)+a^2(1-c\beta)+4a)\xi^2 +2(1+a)\xi+1\Bigg).
\end{split}$$ In particular, the 