Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function. Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders of $t$
$$f(t)\sim\sum_{n=0}^{\infty}c_{i_n}t^{c_{i_n}}\quad\quad t\rightarrow 0$$
where $c_{i_n}$ is a increasing sequence of naturals.
I saw this on an article and have no clue what it means. Specifically what does $\sim$ is in this case?
- Can it mean $f(t)-\sum_{n=0}^{\infty}c_{i_n}t^{c_{i_n}}= \mathcal{o}(g)$ where $g$ is a non analytical smooth functions that is strictly bounded by any polynomial? Like $g(x)=1-e^{1/x}$?
Any clarification would be deeply appreciated. PS: There is really nothing else in the article that explains this better so I am hoping its some common knowledge that I'm not aware of.