Asymptotic expansion

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 $O(z^{N+1})$.