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  ∫ ba f(x)dx by suitable variable transformations x = ø(u). These formulas are characterized by having double exponential asymptotic behavior of the integrands in the resulting infinite integrals as u → ± ∞, 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.