Gauss quadrature rules for finite part integrals

We construct a set of polynomials φn(x,ζ) which are orthogonal with respect to w(x)/(x − ζ)2, where wx is a weight function. These polynomials can be used for the definition of a Gauss quadrature formula for the finite part integral The process is exactly the same as the one used for the extraction of the classical Gauss formula for the Riemann integrals. Three different methods are derived. The first and most accurate quadrature formula is successfully tested in some numerical examples. The proposed quadrature formulas have many applications in problems of mathematical physics, mechanics, etc.