Estimation of errors in the numerical quadrature of analytic functions

Abstract

An estimate of errors in the numerical quadrature of analytic functions is given in connection with the behavior of the integrand in the complex plane.The Integral is transformed into a contour integral as well as its approximation so that the problem of approximation of the integral is reduced to the one of approximation of a transcendental function by a rational function G _n (z)/F _n (z). The function which characterizes the error of the numerical quadrature is introduced and the contour maps of in the complex plane are given for several quadurature formulas. By the use of these maps, investigating the analytic behavior of the integrand in the complex plane, one can know which formula is best suited to the given integrand in the prescribed precision.