# 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 ∫ 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.