# A (simple?) matter of notation

I am currently working on some power series problems, so I deal with the sequence of their (generally complex) coefficients $a_i$, $i\in\mathbb{N}$, denoted in the sequel as $\langle a_n\rangle$ following reference . A sequence is simply a function $$\mathbb{N}\ni n\mapsto a_n\in\mathbb{C}\tag{1}\label{1}$$ Given a finite/infinite subset $S$ of the range of $\langle a_n\rangle$, I need to analyze the indexes $n$ of each member $a_n\in S$. And now my question arises: does there exists a standard notation for the inverse image of $S$ under the function expressed by \eqref{1}?

Having read many textbooks on real and complex analysis, power series and so on, I did not find anything on such matter: however, perhaps specialists in combinatorics or other similar field are more customary with such concepts so I decided to ask this question in order to not introduce in my work useless notations.

 Emanuel Fisher (1983), "Intermediate Real Analysis", Springer Verlag.

• $a^{-1}_S$ or $a^{-1}(S)$ usually
– ℋolo
Sep 19 '18 at 6:09
• How about $\langle a_n \rangle^{-1}(S)$? $\langle a_n \rangle$ denotes a function, so the pre-image notation seems suitable here.
– user445909
Jan 16 '19 at 21:13
• @E-mu: nice comment. Why don't you post it as an answer? Jan 16 '19 at 21:15
• This post is slightly dated and also has an accepted answer, so I am a bit tentative about posting an answer :-). I am also not sure that this notation is standard.
– user445909
Jan 16 '19 at 21:27
• @E-mu: well, even if I accepted Fred's answer, I can upvote yours too. However, if you fear to be downvoted by other members, I'll respect your choice. Jan 16 '19 at 21:31

Denote the function in (1) by $a$, hence $a: \mathbb N \to \mathbb C$ and $a(n)=a_n$.
Then $\{n \in \mathbb N: a_n \in S\}=a^{-1}(S)$.
If we use the notation $$\langle a_n \rangle$$ to denote a sequence $$a$$, where for each $$n \in \mathbb{N}$$, $$a_n := a(n)$$, it seems fitting to write the pre-image of $$S \subseteq \mathbb{C}$$, $$\ a^{-1}(S),$$ as $$\langle a_n \rangle ^{-1}(S).$$
Going from what you mentioned under Fred's post, one advantage of this notation is that we are not hiding the fact that $$a$$ is a sequence. I am not sure whether writing the pre-image this way is standard, however.