Improved error estimates for an exponentially convergent quadratures

Physicists often need to calculate integrals numerically, and with high accuracy. In recent years, it has been shown that for some practically important classes of functions, it is possible to dramatically increase the accuracy and reduce the complexity of quadratures. The paper describes the corresponding mathematical apparatus with the latest improvements, which reduce the complexity of calculations by hundreds of times or more. Examples of physical problems to which it is well applicable are given.