Double exponential formulas for numerical indefinite integration

Abstract

AbstractIn this paper we derive a formula for indefinite integration of analytic functions over (−1,s) where −1<s<1, by means of the double exponential transformation and the Sinc method. The integrand must be analytic on −1<x<1 but may have a singularity at the end points x=±1. The error of the formula behaves approximately as exp(−c1N/logc2N) where N is the number of function evaluations of the integrand. This error term shows a much faster convergence to zero when N becomes large than that of the known formula by Haber. Also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over (0,s), 0<s<∞ and over (−∞,s), −∞<s<+∞. Several numerical examples indicate high efficiency of the formulas