The Gauss–Laguerre quadrature rule for finite‐part integrals

Abstract

AbstractThe classical Gauss–Laguerre quadrature rule for the semi‐infinite integration interval [0,∞] is modified and applied to the case of the weight function exp(‐x)/x corresponding to finite‐part (or, equivalently, hypersingular) integrals. The new set of orthogonal polynomials is constructed and it is seen to consist of linear combinations of the classical Laguerre polynomials with appropriately determined coefficients. The zeros of these modified Laguerre polynomials are seen to be distinct, but one of these lies outside the integration interval. Formulae for the corresponding weights are also given and numerical values and results are presented. The present results generalize the corresponding results for the finite interval [0,1] to semi‐infinite intervals and they are applicable to a variety of applied mechanics and related problems, where finite‐part integrals appear in a natural way.