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In this paper, we consider a quadrature rule for Cauchy integrals of the form I(wf ; s)= 1 &1 w(t) f (t)Â(t&s) dt, &1 &1Â2. Using the change of variables t=cos y, s=cos x and subtracting out the singularity, we propose a trigonometric quadrature rule. We obtain the error bounds independent of the set of values of poles and construct an automatic quadrature of nonadaptive type. AcademicPress

SUMMARYTwo trigonometric quadrature formulae, one of non-interpolatory type and one of interpolatory type for computing the hypersingular integral =2 d are developed on the basis of trigonometric quadrature formulae for Cauchy principal value integrals. The formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae. Fast three-term recurrence relations for evaluating the quadrature weights are derived. Numerical tests are carried out using the current formula. As applications, two simple crack problems are considered. One is a semi-inÿnite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions. It is found that the present method generally gives superior results.

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This paper is concerned with a Chebyshev quadrature rule for approximating one sided finite part integrals with smooth density functions. Our quadrature rule is based on the Chebyshev interpolation polynomial with the zeros of the Chebyshev polynomial T N+1 ({)&T N&1 (t). We analyze the stability and the convergence for the quadrature rule with a differentiable function. Also we show that the quadrature rule has an exponential convergence when the density function is analytic. 2001Academic Press