A class of quadrature formulas of Chebyshev type for singular integrals (Q1059982)

We reprove and extend several quadrature formulas for singular integrals \(\int^{1}_{-1}g(t)/(t-x)(1-t^ 2)^{1/2}dt\quad (-1<x<1)\) or \(\int^{1}_{-1}g(x)(1-x^ 2)^{1/2}/(x-t)dx\quad (-1\leq t\leq 1)\) systematically by a unified method based on the corresponding formulas for ordinary integrals so that their remainders can be easily estimated. Using the same method, we also establish another set of formulas named Simpson-Chebyshev type which seems more effective when the density function g(x) has less order of smoothness as illustrated by an example.