Some Derivative-Free Quadrature Rules for Numerical Approximations of Cauchy Principal Value of Integrals

Some derivative-free six-point quadrature rules for approximate evaluation of Cauchy principal value of integrals have been constructed in this paper. Rules are numerically verified by suitable integrals, their degrees of precision have been determined, and their respective errors have been asymptotically estimated. 1. Introduction Recently Das and Hota [1] have constructed a derivative-free 8-point quadrature rule for numerical evaluation of complex Cauchy Principal Value of integrals of type along the directed line segment , from the point to the point , and is assumed to be an analytic function in a domain containing . The objective of this paper is to obtain some other quadrature rules having six-nodes not involving derivative of the function for numerical approximation of the complex CPV integrals given in (1) from the family of rules given by Das and Hota [1]. 2. Formulation of Rules Das and Hota [1] have given the following derivative-free 8-point parametric quadrature rule of degree of precision at most ten to approximate the integrals of the type given in (1): where the rule given in (2) is of precision eight for . However, the rule may be reduced to a six point rule for suitable values of the parameter “ .” without altering its algebraic degree of precision that is eight. These rules are as follows:(i) ; for this value of ; the weight in (3) is zero and the rule given in (2) becomes a six point rule denoted by given as: (ii) ; in this case the weight and the corresponding rule (denoted by ) is: (iii) ; as in the case of two cases noted above, we found here that the weight for this value of ; and the rule denoted by becomes: Each of these rules, that is, , and is a six-point rule. For the numerical integration of the integral (1) it is required to evaluate the function at six points instead of eight points as in the case, the rule proposed by Das and Hota [1]. Both the rules and have as nodes and hence they are closed type of rules. It is pertinent to note here that the degree of precision of each of the rules , , and is eight which is the same as that of the rule except for the value of in which case it becomes a rule of precision ten; however, in this case, evaluation of function at 8 nodes is required in approximation of integrals. 3. Error Analysis The error associated with the rule as given in (2) is We assume here that the function is analytic in the disc under this assumption, can be expanded in terms of the Taylor’s series about the point in the disc as Where are the Taylor’s coefficients. As the series given in (9) is uniformly