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.
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Hidetosi Takahasi, Masatake Mori, Double Exponential Formulas for Numerical Integration. Publ. Res. Inst. Math. Sci. 9 (1973), no. 3, pp. 721–741DOI 10.2977/PRIMS/1195192451