Analytic theory of finite asymptotic expansions in the real domain. Part I: Two-term expansions of differentiable functions (Q452845)

The author studies a general analytic theory of asymptotic expansion of functions \[ \phi(\xi)= \alpha_1 \phi_1(\xi)+\dots +\alpha_\nu\phi_\nu(\xi)+ o(\phi_\nu(\xi)), \;\xi\to \xi_0,\;\nu\geq 2, \] where the comparison functions \(f_i\) are supposed to be known in a neighbourhood of \(\xi_0\) and form an asymptotic scale at \(\xi_0\), i.e., \[ \phi_1\gg\phi_2(\xi)\gg\dots \gg\phi_\nu(\xi),\;\xi\to\xi_0. \] The author deals with real-valued functions of one real variable. The simplest and first historical example of the last relation, as all of us know, is Taylor's formula for which the author has at disposal not only the elementary rules for manipulating the corresponding asymptotic relations but also the various forms of Taylor's theorem. As far as general asymptotic expansions are concerned, the current state of affairs is that we are able to perform practical manipulations.