Recall:
Theorem 1 (Cauchy's Integral Formula for Derivatives): Let $A \subseteq \mathbb{C}$ be open and let $f: A \rightarrow \mathbb{C}$ be analytic on $A$. Let $\gamma$ be any simple closed piecewise smooth and positively oriented curve contained in $A$ and such that the inside of $\gamma$ is contained in $A$. Then if $z_{0}$ is inside $\gamma$ we have that for all $k=0,1,2, \ldots$ that $$f^{(k)}\left(z_{0}\right)=\frac{k !}{2 \pi i} \int_{\gamma} \frac{f(z)}{\left(z-z_{0}\right)^{k+1}} d z$$
There exist some version to this theorem to $\mathbb{C}_\infty$? I am interested to prove the following function is $C^{\infty}$ class arround the origin: Let $g \colon \mathbb{C}_{\infty} \to \mathbb{C}_{\infty}$ $$g(w)=\Psi^{-1}w+\Psi^{-1}w^{2}r+\Psi^{-1}w^{3}r^{2}+\Psi^{-1}w^{4}r^{3}+\ldots$$ With $\Psi \colon \mathbb{C}_{\infty} \to \mathbb{C}_{\infty}$ is an isometry, and I need some condition over $r$ to ensure that $g$ is $C^\infty$ class, my attempt if there exist some version of Cauchy integral formula to extended Complex plane I need the condition over $g$ to be analytic.