Double Exponential Formulas for Numerical Integration

A family of numerical quadrature formulas is introduced by application of the trapezoidal rule to infinite integrals which result from the given integrals f b \ f(x)dx by suitable variable transformations x = <j)(u}. These formulas are characterized by having double exponential asymptotic behavior of the integrands in the resulting infinite integrals as ii-»±oo, 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. §