An IMT-Type Double Exponential Formula for Numerical Integration

Abstract

A quadrature formula for evaluation of improper integrals over $(-1,1)$ is presented, which is obtained in such a way that the interval of integration $(-1,1)$ is transformed into $(-\infty ,\infty )$ by $x=\tanh (A\sinh 2Bu/(1-u^2))$ and that the trapezoidal rule with an equal mesh size is subsequently applied to the transformed integral. Asymptotic error analysis is made by means of the method of contour integral and a comparison with the IMT-rule and the double exponential formula is given along with some numerical examples.