The Double Exponential Formulas for Numerical Integration over the Half Infinite Interval

Abstract

It is known that a class of quadrature formulas called the double exponential formulas obtained by variable transformation are very efficient for numerical integration of an analytic function over a finite interval, in particular when it has some end-point singularity. It is also useful for integration over an infinite interval. In this paper the double exponential formulas for integrals of several types over the half infinite interval are discussed and it is shown how to arrange them into an automatic non-adaptive subroutine. A modification of the double exponential formula suitable for integration of slowly decaying oscillatory functions using the Richardson extrapolation technique is given. Also contour maps of the characteristic functions of the error of the double exponential formulas over the half infinite interval are shown which may be helpful when estimating the error of the formulas applied to a specific integral.