The Euler-Maclaurin formula for functions with singularities
Asymptotic formulas for the sum \((1/n)\sum^{n- 1}_{k= 1}\Phi(k/n)\) as \(n\to\infty\) are obtained, where \(\Phi\) is a function of the form \(\Phi(x)= x^{-\alpha_1}(\log x)^{d_1}(1- x)^{-\alpha_2}(\log(1- x))^{d_2}\phi(x)\), and \(\phi(x)\) is sufficiently smooth on \([0, 1]\).
scientific article; zbMATH DE number 1059799