Asymptotic estimation of Gaussian quadrature error for a nonsingular integral in potential theory

This paper considers the n-point Gauss-Jacobi approximation of nonsingular integrals of the form ∫ 1 −1 μ(t)φ(t) log(z− t) dt, with Jacobi weight μ and polynomial φ, and derives an estimate for the quadrature error that is asymptotic as n → ∞. The approach follows that previously described by Donaldson and Elliott. A numerical example illustrating the accuracy of the asymptotic estimate is presented. The extension of the theory to similar integrals defined on more general analytic arcs is outlined.