Quadrature formulas for singular integrals on a fixed-node integration segment

Sh. S. Khubezhty^abc, L. Yu. Plieva<sup>c

a</sup> Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz
^b Financial University under the Government of the Russian Federation, Vladikavkaz Branch
^c North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz

Abstract: Quadrature formulas are constructed for singular integrals on the integration segment [-1, 1], with weight functions (t)=1/√(1-t^2 ), p(t)=√(1-t^2 ). The construction uses the Lagrange interpolation polynomial with function values at fixed points {-1; 1} and at zeros of Chebyshev polynomials orthogonal to [-1, 1] with respect to the corresponding weight functions. When calculating the coefficients of the quadrature formula, formulas for singular integrals are used, where Chebyshev polynomials of the first and second genera are taken as the density. The formulas obtained are quadrature formulas of the interpolation type. The paper provides examples confirming the effectiveness of the obtained quadrature formulas for calculating singular integrals. In frequency, the functions φ(t)=1,φ(t)=t and φ(t)=t^2 are used as density, for which accurate results are obtained. Estimates of the error are given for the obtained quadrature formulas. The resulting quadrature formulas have an algebraic degree of accuracy of n+1.

UDC: 519.64

Received: 26.06.2025
Accepted: 11.11.2025

DOI: 10.18522/1026-2237-2025-4-20-24

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