Abstract

Quadrature formulas are presented for integrals with a logarithmic singularity and with a sign function, containing the weight function of Jacobi orthogonal polynomials, the exponents of which can be complex numbers with a real part greater than minus one. The latter are remarkable in that they have the same structure as the quadrature formulas for singular and regular integrals and can be used to solve singular integral equations that also contain terms with a weak singularity. Formulas for calculating the integral with a logarithm at an arbitrary point of the complex plane are also presented, and by numerical analysis, the area around the interval is outlined, outside of which this integral, with a certain degree of accuracy, can also be calculated using the quadrature formula for smooth functions.

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