# Double Exponential Formulas for Numerical Integration

### Hidetosi Takahasi

University of Tokyo, Japan### Masatake Mori

Tokyo Denki University, Japan

## Abstract

A family of numerical quadrature formulas is introduced by application of the trapezoidal rule to infinite integrals which result from the given integrals $∫_{a}f(x)dx$ by suitable variable transformations $x=ϕ(u)$. 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.

## Cite this article

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

DOI 10.2977/PRIMS/1195192451