  This paper concerns an analytic and numerical analysis of a class of weighted
singular Cauchy integrals with exponential weights $w:=\exp(-Q)$ with finite
moments and with smooth external fields $Q:\mathbb R\to [0,\infty)$, with
varying smooth convex rate of increase for large argument. Our analysis relies
in part on weighted polynomial interpolation at the zeros of orthonormal
polynomials with respect to $w^2$. We also study bounds for the first
derivatives of a class of functions of the second kind for $w^2$.
