# Publications of the Research Institute for Mathematical Sciences

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**Volume 9, Issue 3, 1973, pp. 721–741**

**DOI: 10.2977/prims/1195192451**

Double Exponential Formulas for Numerical Integration

Hidetosi Takahasi and Masatake Mori^{[1]}(1) Department of Mathematical Sciences, Tokyo Denki University, Hatoyama, Hiki-gun, 350-0394, TOKYO, JAPAN

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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Takahasi H, Mori M. Double Exponential Formulas for Numerical Integration. *Publ. Res. Inst. Math. Sci.* 9 (1973), 721-741. doi: 10.2977/prims/1195192451