Yet another DE-Sinc indefinite integration formula
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Based on the Sinc approximation combined with the tanh transformation, Haber derived an approximation formula for numerical indefinite integration over the finite interval (-1, 1). The formula, (SE1) uses a special function for the basis functions. In contrast, Stenger derived another formula (SE2), which does not use any special function but does include a double sum. Subsequently, Muhammad and Mori proposed the formula (DE1), which replaces the tanh transformation with the double-exponential transformation in the formula (SE1). Almost simultaneously, Tanaka et al. proposed the formula (DE2), which was based on the same replacement in (SE2). As they reported, the replacement drastically improves the convergence rate of (SE1) and (SE2). In addition to the formulas above, Stenger derived yet another indefinite integration formula (SE3) based on the Sinc approximation combined with the tanh transformation, which has an elegant matrix-vector form. In this paper, we propose the replacement of the tanh transformation with the double-exponential transformation in the formula (SE3). We provide a theoretical analysis as well as a numerical comparison.
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