Asymptotic assessments are widely used in various branches of mathematics. For some of them, asymptotic methods are described in sufficient detail in works of a monographic nature, for example, in the theory of ordinary differential equations and number theory. In recent decades, methods of asymptotic decomposition of integrals have been intensively developing, however, in works devoted to the asymptotic representation of functions, the latest achievements in this direction are not fully reflected, particular results are scattered in journal articles and there are cases when the same method is proposed by different authors.