Asymptotic scale

Summary

An asymptotic scale, also called an asymptotic sequence, is a sequence of functions $\varphi_0, \varphi_1, \dots$ near a point $x_0$. Each function in it is negligible compared with the one before it. It is the system of reference functions in which asymptotic expansions are written. The most common scale is the powers $x^{-n}$ as $x \to \infty$, but scales with logarithms, fractional powers or exponentials are also needed. Fixing a scale makes the coefficients of an expansion unique.