(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the punctured complex plane, not every formal solution to a differential equation $Pf = 0$ is convergent. Additionally, even if one has access to a convergent solution, it is often better (algebraically, analytically, or geometrically) to work with its asymptotic expansions -- if you remember slogging through various lectures on special functions in elementary calculus, these are the beasts to keep in mind.
Even some easy functions-- like $f(x) = \mathrm{exp}(-1/x)$, the fundamental solution to the algebraic differential equation $(x^2\frac{d}{dx}+1)f = 0$-- are asymptotic to zero as $x \to 0$ when $\mathrm{Re}(x) > 0$, but blow-up faster than any algebraic meromorphic function for $x \to 0$ when $\mathrm{Re}(x) < 0$. Consequently, the best approach when dealing with such functions is to consider their asymptotic behavior on different arc-sectors about $0$, where by an arc-sector, I mean a subset of $\mathbb{C}^*$ of the form $$ S_r(a,b) := \{ x \in \mathbb{C} \, | \, 0 < |x|<r, a < \mathrm{arg}(x) < b\} $$ for $r > 0$ and $a < b$ two real numbers (usually of width less than $2\pi$). An asymptotic expansion at $0$ of a function $f \in \Gamma(S_r(a,b);\mathcal{O}_\mathbb{C})$ is a formal complex power series $\hat{f} = \sum_{n \geq 0} c_n x^n \in \mathbb{C}[[x]]$ such that, for all $N \in \mathbb{N}$, and every closed subsector $W \subset S_r(a,b)$, there is a constant $C(N,W) > 0$ such that the estimate $$ \big | f(x) - \sum_{n = 0}^{N-1}c_n x^n \big | \leq C(N,W)|x|^N $$ holds for all $x \in W$. If such an $\hat{f}$ exists, it is unique. Let $\mathcal{A}(S_r(a,b))$ denote the subset of all such holomorphic functions with asymptotic expansions on $S_r(a,b)$, and write \begin{align*} J : \mathcal{A}(S_r(a,b)) &\to \mathbb{C}[[x]] \\ f &\mapsto \hat{f} \end{align*} for the "Taylor map" sending a function to its asymptotic expansion. To avoid dealing with issues like choosing $r>0$ small enough, we take the direct limit $\underset{r>0}{\varinjlim}\mathcal{A}(S_r(a,b)) =: \mathcal{A}(S(a,b))$ and thus obtain a sheaf $\mathcal{A}$ of differential $\mathbb{C}$-algebras on $S^1$ consisting of holomorphic functions with asymptotic expansions at $0$. A foundational result in this context is the
Borel-Ritt lemma:
for any open interval $(a,b) \neq S^1$, the map $J: \mathcal{A}(S(a,b)) \to \mathbb{C}[[x]]$ is surjective.
If we let $\mathbb{C}[[x]]$ denote the constant sheaf with stalk $\mathbb{C}[[x]]$ on $S^1$, we obtain a short exact sequence of sheaves on $S^1$ $$ 0 \to \mathcal{A}^0 \to \mathcal{A} \xrightarrow{J} \mathbb{C}[[x]] \to 0 $$ where $\mathcal{A}^0$ denotes the sheaf of holomorphic functions which are asymptotic to $0$ (e.g., our friend $\mathrm{exp}(-1/x)$ from earlier, on any sector contained in the right halfplane).
My question: Is the Taylor map $J$ continuous on any proper open sector $S(a,b)$?
$\mathcal{A}(S(a,b))$ is a Fréchet space, and $\mathbb{C}[[x]]$, with its canonical $(x)$-adic topology, is a (non-archimedean) Banach space. Additionally, $\mathcal{A}^0(S(a,b))$ is a nuclear Fréchet space. To me, the criterion for $f$ to have $\hat{f}$ as its asymptotic expansion looks very much like the definition of convergence in the $(x)$-adic topology as well.
Let $S = S(a,b)$. I believe this question can be reduced to that of continuity at $0$ in $\mathcal{A}(S)$, in which case it takes the following form: let $\{f_n\}_n \subset \mathcal{A}(S)$ be a family such that $f_n \xrightarrow{n \to \infty} 0$. Then, for all $j \in \mathbb{N}$, there exists a $N \in \mathbb{N}$ such that for all $n \geq N$, we have $$ \underset{x \to 0, x \in S}{\lim} x^{-k}f_n(x) = 0 $$ for all $0 \leq k < j$. However, I don't know of a specific family of seminorms which generate the topology on $\mathcal{A}(S)$. Consequently, I don't know how to use the assumption that the sequence $\{f_n\}_n$ converges to $0$ in $\mathcal{A}(S),$ in order to control their growth rates against the $x^{-k}$.
Any help would be appreciated, especially if it's to tell me this claim is obviously false!
