Double Exponential Formulas for Numerical Integration

Abstract

A family of numerical quadrature formulas is introduced by application of the trapezoidal rule to infinite integrals which result from the given integrals  ${\int }_a^{b}f(x)dx$ by suitable variable transformations $x=\varphi (u)$. These formulas are characterized by having double exponential asymptotic behavior of the integrands in the resulting infinite integrals as $u\to \pm \infty$, and it is shown both analytically and numerically that such formulas are generally optimal with respect to the ecomony of the number of sampling points.