content_image/312703.jpg content_image/312704.jpg content_image/312705.jpg content_image/312706.jpg content_image/312707.jpg
where 

<img src='content_image/312703.jpg'>

<img src='content_image/312704.jpg'>

<img src='content_image/312705.jpg'>

and the contours of integration over m$_{1}$ and m$_{2}$ in (58-60) are obtained from the contours of integration over z$_{1}$ and z$_{2}$ in (54-56) by transformation m ( z ) . Recall that by assumption the contours over z$_{1}$ and z$_{2}$ intersect the real line to the left of zero and in between the upper boundary of the support of the Marchenko-Pastur $_{distribution, (1 +}$√$_{c}$$_{)}$ $^{2}$, and ¯ z$_{j}$$_{0}$. Therefore, as can be shown using the definition (57) of m ( z ), the m$_{1}$ -contour and m$_{2}$ -contour are clockwise oriented and intersect the real line in between − $_{(1 +}$√$_{c}$$_{)}$ − 1 and m (¯ z$_{j}$$_{0}$ ) = − h$_{j}$ ( h$_{j}$ + c ) − 1 and to the right of zero. In particular, both contours enclose 0 and − h$_{j}$ ( h$_{j}$ + c ) − $^{1}$, but not − 1 and − (1 + h$_{j}$ ) − $^{1}$. 

Assuming without loss of generality that m$_{1}$ -contour lies inside the m$_{2}$ -contour, from (A64) in the Supplementary appendix of OMH, we have 

<img src='content_image/312706.jpg'>

Denoting − h$_{j}$ ( h$_{j}$ + c ) − 1 as x$_{j}$, we get from (62) and (58) 

<img src='content_image/312707.jpg'>

where the last equality follows from Cauchy’s residue theorem and the fact that 

