Numer. Math. 46, 121-130 (1985) Numerische Mal ematik 9 Springer-Verlag1985 On the Evaluation of Certain Two-Dimensional Singular Integrals with Cauchy Kernels* C. Dagnino and A. Palamara Orsi Dipartimento di Matematica, C. Duca degli Abruzzi, 24, 1-10129 Torino, Italy Summary. In this paper we consider product formulas of interpolatory type for the two-dimensional Cauchy principal value integral: 1 1 f(x, y) I(f;t/1,t/2)= ~ ~ ~o(x,y) (x dxdy -I -1 --r/1) (Y --r/2) where: r/l, t/2e(--1 , 1), o~(x,y)=tol(x).fo2(y ) and (DI(X) and ~2(Y) are two absolutely integrable weight functions. The integral is approximated by i=0 j=O where the nodes {xl} and {yj} are the zeros of the Chebyshev polynomials of the first kind, commonly named "classical" abscissas, or the Clenshaw points, often called "practical" abscissas. We present convergence results for these rules. Subject Classifications: AMS (MOS): 65D32, 41A17; CR: G 1.4. 1. Introduction F o r t/1 , ~/2G( - 1, 1) let I(f;th,/~2) denote the two-dimensional Cauchy principal value integral: 1 f(x, y) I(f;/~1,/~2) = ~ ~_1o~(x,Y) (x dxdy (1) -1 --~/1) (y --r/2) where e)(x, y) is of the form: co(x, y) = co 1(x). e~2(Y) * Work sponsored by the "Ministero della Pubblica Istruzione" of Italy