On numerical integration methods with the generalized Stieltjes weight function

A class of weighted quadrature rules whose weight function is denoted by x(a+b ln x), x ∈ [0, ∞) is introduced and investigated. The integration formulas, presented in this paper, generally obey the following form:∫0∞x(a+blnx)f(x)dx=∑i=1nwif(xi)+Rn[f],where xi are the zeros of orthogonal polynomials with respect to the introduced weight function ; wi the related coefficients and Rn[f] as the integration error function. In this way, some analytic and numerical examples are also given.