Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates

Let ∏n−1[f] be the polynomial of degree n−1 interpolating the function f at the points x1,x2, …,xn with Pn(xi) = 0, i.e., at the nodes of the classical Gaussian quadrature formula. For the numerical approximation of the Cauchy principal value integral ⨍1−1 ƒ(x)(x−λ)−1 dx with λ ∈ (−1,1) and f ∈ C1[−1,1], we present the quadrature formula Qn+1G3 given by Qn+1G3[ƒ;λ]:=∫−1πn−1[ƒ](x)−πn−1[ƒ](λ)x−λdx+ƒ(λ)1n1−λ1+λ. w that this quadrature formula does not have the disadvantages of the other two well-known quadrature formulae based on the same set of nodes. In particular, we prove that the sequence ( based on the same set of nodes. In particular, we prove that the sequence (Qn + 1G3[f; λ]) converges to the true value of the integral uniformly for all λ ∈ (−1, 1). We give estimates for the error term. Furthermore, we state some relations connecting the present quadrature formula to the previously introduced formulae.