Asymptotic analysis

Definition

Let $D \subseteq \R$ and let $x_0$ be a limit point of $D$, possibly $\pm\infty$. A sequence of functions $\varphi_0, \varphi_1, \varphi_2, \ldots$ on $D$ is an asymptotic sequence as $x \to x_0$ if $\varphi_{n+1}(x) = o(\varphi_n(x))$ as $x \to x_0$ for every $n$. Let $f$ be a function on $D$ and let $a_0, a_1, \ldots$ be numbers. The formal series $\sum_{n \ge 0} a_n \varphi_n$ is an asymptotic expansion of $f$ as $x \to x_0$, written $$f(x) \sim \sum_{n=0}^{\infty} a_n \varphi_n(x) \quad (x \to x_0),$$ if for every $N \ge 0$, $$f(x) - \sum_{n=0}^{N} a_n \varphi_n(x) = o(\varphi_N(x)) \quad (x \to x_0).$$ Asymptotic analysis is the study of such expansions and of asymptotic equivalence $f \sim g$, meaning $f / g \to 1$. Its estimates are written with big O and little o notation.