Quadrature formulas for Cauchy-type integrals with the Cauchy kernel to a integer power and a Jacobi weight function with complex exponents

Abstract

The paper presents quadrature formulas for hypersingular integrals of various orders. It is assumed that the density of these integrals is represented as a product of a function that satisfies the Hölder condition and a weight function of Jacobi orthogonal polynomials. In this case, the exponents of the weight function can be complex numbers, the real part of which is greater than -1. Numerical analysis of the dependence of the root-mean-square deviation of the quadrature formula of order 8 on the value of the hypersingular integral calculated using standard software packages is carried out for various complex values of the weight function exponents. For hypersingular integrals up to the fourth order inclusive, a numerical analysis of the convergence of quadrature formulas is carried out for certain complex values of the exponents of the weight function.