Double exponential formulas for numerical indefinite integration

Authors:
Mayinur Muhammad;Masatake Mori
Affiliations:
Department of Mathematical Sciences, Tokyo Denki University, Hatoyama-cho, Hiki-gun, Saitama 350-0394, Japan;Department of Mathematical Sciences, Tokyo Denki University, Hatoyama-cho, Hiki-gun, Saitama 350-0394, Japan
Venue:
Journal of Computational and Applied Mathematics
Year:
2003

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Abstract

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 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.