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Summary. A quadrature rule is described for the numerical evaluation ofHadamard finite-part integrals with a double pole singularity within the range of integration. The rule is based upon the observation that such an integral is the derivative of a Cauchy principal value integral.

Abstract.Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, arediscussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integral, are proved.

Abstract.An algorithm, based on the use of orthogonal polynomials, for product-integration is outlined. A general discussion on the convergence of such quadrature rules for finite intervals is then given. The paper concludes with five examples for each of which sufficient conditions for convergence of the quadrature rule are given.

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