Analytic and Numerical Analysis of Singular Cauchy integrals with exponential-type weights

Let $I=(c,d)$, $c < 0 < d$, $Q\in C^1: I\rightarrow[0,\infty)$ be a function with given regularity behavior on $I$. Write $w:=\exp(-Q)$ on $I$ and assume that $\int_I x^nw^2(x)dx<\infty$ for all $n=0,1,2,\ldots$. For $x\in I$, we consider the problem of the analytic and numerical approximation of the Cauchy principal value integral: \begin{equation*} I[f;x]:=\lim_{\varepsilon \to 0+} \left( \int_{c}^{x-\varepsilon} w^2(t)\frac{f(t)}{t-x}dt+ \int_{x+\varepsilon}^{d} w^2(t)\frac{f(t)}{t-x}dt. \right) \end{equation*} for a class of functions $f: I\rightarrow \mathbb{R^+}$ for which $I[f;x]$ is finite. In [1-4], the first two authors studied this problem and some of its applications for even exponential weights $w$ on $(-\infty,\infty)$ of smooth polynomial decay at $\pm \infty$ and given regularity.