Improper integrals, simple integrals, and numerical quadrature

Abstract

x is defined in the simplest case as a limit of proper Riemann integrals, which are themselves limits of Riemann sums. In this paper we discuss the representation of the improper integral as a limit of Riemann sums. Our main result-Theorem 3-gives a condition on the integrand that is necessary and sufficient for such represen- tation of the integral, with the largest natural class of Riemann sums. Some of the motivation for this paper comes from the theory of numerical integration. Most formulas for numerical quadrature-Simpson’s rule, the trapezoid rule, and the Gauss-Legendre formulas, for example -approximate the integral by calculating carefully chosen Riemann sums.* Thus it is the Riemann concept of the integral that is most appropriate for numerical analysis. The quadrature rules mentioned converge to the integral whenever the function being integrated is properly Riemann-integrable; there seems to be no larger class of bounded functions for which quadrature rules converge. In the case of the improper Riemann integral, the connection with numeri- cal quadrature is obscured by the double limiting process involved. If we wish to use a sequence of quadrature formulas, or quadrature rule, (with