Double exponential formulas for numerical indefinite integration

In 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 Sine 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(-c(1)N/log c(2)N) 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 < infinity and over (-infinity,s), -infinity < s < + infinity. Several numerical examples indicate high efficiency of the formulas. (C) 2003 Elsevier B.V. All rights reserved.