The double-exponential transformation in numerical analysis

The double-exponential transformation was first proposed by Takahasi and Mori in 1974 for the efficient evaluation of integrals of an analytic function with end-point singularity. Afterwards, this transformation was improved for the evaluation of oscillatory functions like Fourier integrals. Recently, it turned out that the double-exponential transformation is useful not only for numerical integration but also for various kinds of Sine numerical methods. The purpose of the present paper is to review the double-exponential transformation in numerical integration and in a variety of Sine numerical methods. (C) 2001 Elsevier Science B.V. All rights reserved. MSC. 65D30; 65D32.