A Unified Approach to Quadrature Rules with Asymptotic Estimates of Their Remainders

The starting point of this paper is a theorem, based on the theory of functions of a complex variable, which essentially gives a representation of the remainder in a quadrature rule as a contour integral. From this theorem it is shown how most of the well-known interpolatory and non-interpolatory quadrature rules may be derived. Some new quadrature rules are also given. The latter part of the paper is devoted to the evaluation of asymptotic estimates of the remainder, by consideration of particular examples. It is shown how this asymptotic analysis frequently gives excellent estimates of the remainder.