In going through Folland's Abstract Harmonic Analysis, I came on the following. Let $\mathcal{A}$ be a unital Banach algebra with unit $e$, and define $$\sigma(x) = \{ \lambda \in \mathbb{C} : \lambda e - x \textrm{ is not invertible} \}$$ to be the spectrum of $x \in \mathcal{A}$. The spectrum of $x$ will be compact. Define $R : \mathbb{C} \setminus \sigma (x) \to \mathcal{A}$ by $R(\lambda) = (\lambda e - x)^{-1}$. The author claims this function is "analytic", which he proves (Lemma 1.5) by observing that if $\lambda, \mu \in \mathbb{C} \setminus \sigma(x)$, then $$\lim_{\mu \to \lambda} \frac{R(\mu) - R(\lambda)}{\mu - \lambda} = - R(\lambda)^2 .$$ Therefore this is a complex-differentiable function from $\mathbb{C} \setminus \sigma(X)$ to $\mathcal{A}$.
My question is: What does it mean for an $\mathcal{A}$-valued function to be "analytic"? I see that he proves this fact by showing that it's complex-differentiable, but my understanding of the concepts was that (in the elementary case of $\mathbb{C}$-valued functions) complex-differentiable meant what it sounds like, analytic meant being locally expressed as a power series, and it was a theorem of complex analysis that these concepts were equivalent in the context of functions from open subsets of $\mathbb{C}$ to $\mathbb{C}$. However, I'm unfamiliar with the idea of analytic functions in this context, as well as how this concept proves equivalent to complex differentiability in this setting.
Thanks!