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Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate a double Cauchy singular integral where the density function is enough smooth (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70, 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper

A note on a paper by G. Mastroianni and G. Monegato