Remarks on the gaussian quadrature rule for finite-part integrals with a second-order singularity

Abstract

The Gaussian quadrature rule for finite-part integrals with a second-order singularity (that is, weight function equal to 1/ y 2 on [0,1]), proposed by Kutt, is revisited here. It is proved that the related orthogonal polynomials can be expressed as a linear combination of three successive shifted Legendre polynomials. The coefficients in this linear combination are expressed in an exact, closed, and simple form either by using the nodes and weights of the classical Gauss-Legendre quadrature rule on [0,1] or by using simple finite sums. Formulae for the weights of the relevant quadrature rule are also proposed. Numerical results, derived in numerical experiments, verify also the convergence of the considered numerical integration rule. A large variety of applications of the present results to applied mechanics and engineering is reported in brief, but the main application is to the hypersingular integral equations of crack problems in the theory of two- and three-dimensional elasticity.