Indefinite Integration Formulas Based on the Sinc Expansion

Abstract

The “Tanh rule” of C. Schwartz approximates $\smallint^{1}_{-1}g(u)$ by

$$I_{N}=h\,\underset{j=-N}{\overset{N}{\sum}}\frac{g(tanh\frac{jh}{2})}{2cosh^{2}\frac{jh}{2}}$$

, where h = h(N) is given by a certain formula; $h\!\sim\!cN^{-\frac{1}{2}}$. This integration rule has been shown to be remarkably accurate. Its error is $O(e^{-c\sqrt{N}})$ for some c > 0 even when the integrand is infinite at an endpoint of the integration interval.