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Quadrature formulas obtained by variable transformation

Authors:	Hidetosi Takahasi Masatake Mori

Institution:	(1) Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo, Japan;(2) Research Institute for Mathematical Sciences, Kyoto University, Kitasirakawa, Kyoto, Japan

Abstract:	Quadrature formulas suitable for evaluation of improper integrals such as $\int\limits_{ - 1}^1 {f(x)(1 - x)^{ - \alpha } (1 + x)^{ - \beta } dx,\alpha ,\beta< 1}$ are obtained by means of variable transformations =tanhu and =erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals $\int\limits_\infty ^\infty {f(x)dx}$ by means of variable transformations =sinhu and =tanu.

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