A Comparison Between the Direct and the Classical Numerical Methods for the Solution of Cauchy Type Singular Integral Equations

Cauchy type singular integral equations can be solved numerically either directly (through the use of an appropriate numerical integration rule and reduction to a system of linear equations) or after a previous reduction to an equivalent Fredholm integral equation of the second kind and application of the numerical technique to this equation. In this note it is proved in the special case when the Gauss-Chebyshev numerical integration rule is used that both methods are equivalent in the sense that they provide the same numerical results for the same number of abscissas used in numerical integrations.