A direct method for the construction of gaussian quadrature rules for Cauchy type and finite-part integrals

Abstract

It is shown how the construction of Gaussian quadrature rules for Cauchy type principal value integrals, as well as for finite-part integrals with an algebraic singularity, can be based on the theory of Gaussian quadrature rules for ordinary integrals by using nonclassical weight functions (distributions), but classical systems of orthogonal polynomials. A series of quadrature rules for the aforementioned class of integrals is derived by this new approach and this illustrates its possibilities.

Keywords

• numerical integration
• Gaussian quadrature rules
• Cauchy type integrals
• finite-part integrals
• orthogonal polynomials
• nonclassical weight functions