---
tag: B53THC8F79
kind: definition
name: Asymptotic expansion
msc: [41A60, "30E15", "34E05"]
defines: BJQTEPEZ9G
uses: [04TSHB9H8D]
sources:
  - url: https://en.wikipedia.org/wiki/Asymptotic_expansion
---

## Definition

Let $D$ be a subset of $\R$ or of $\C$, and let $x_0$ be a limit point of $D$, possibly $\infty$.
An *asymptotic sequence* at $x_0$ is a sequence of functions $\varphi_0, \varphi_1, \varphi_2, \ldots$ on $D$ such that
$$\varphi_{n+1}(x) = o(\varphi_n(x)) \quad \text{as } x \to x_0 \text{ in } D$$
for each $n \ge 0$, where $o$ is [[little o notation|04TSHB9H8D]].
Let $f$ be a function on $D$ and let $a_0, a_1, a_2, \ldots$ be complex numbers.
The formal series $\sum_{n \ge 0} a_n \varphi_n$ is an *asymptotic expansion* of $f$ at $x_0$ if for each $N \ge 0$
$$f(x) - \sum_{n=0}^{N} a_n \varphi_n(x) = o(\varphi_N(x)) \quad \text{as } x \to x_0 \text{ in } D.$$
This is written $f(x) \sim \sum_{n \ge 0} a_n \varphi_n(x)$ as $x \to x_0$.

## Remarks

Example: on $D = (1, \infty)$ with $x_0 = \infty$ and $\varphi_n(x) = x^{-n-1}$, one has $\frac{1}{x-1} \sim \sum_{n \ge 0} x^{-n-1}$.
Check: $\frac{1}{x-1} - \sum_{n=0}^{N} x^{-n-1} = \frac{x^{-N-1}}{x-1}$, and this is $o(x^{-N-1})$ because $\frac{1}{x-1} \to 0$.
Non-example: $\varphi_0(x) = x^{-1}$, $\varphi_1(x) = x^{-1}$ is not an asymptotic sequence at $\infty$, because $\varphi_1 = o(\varphi_0)$ fails.
For a fixed asymptotic sequence, the coefficients are unique: $a_N$ is the limit of $\big(f(x) - \sum_{n<N} a_n \varphi_n(x)\big)/\varphi_N(x)$ as $x \to x_0$, whenever $\varphi_N$ does not vanish near $x_0$.
Different functions can have the same expansion: $e^{-x}$ and $0$ both have the zero expansion in powers of $x^{-1}$ as $x \to +\infty$.
The most common case is the power scale $\varphi_n(x) = x^{-n}$ at $\infty$ or $\varphi_n(z) = z^n$ at $0$; an equivalent condition is then that the error after $N$ terms is $O(x^{-N-1})$, respectively $O(z^{N+1})$.