Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally expanded in a power series of $z$?
Take as a concrete example $f_A(z) = \exp(zA-\overline{z}A^*)$ for some fixed $A\in \mathcal{A}$. Is $f_A$ an analytic function?
The holomorphic functional calculus implies that any function that you would "expect" to be analytic is analytic and that the power series is exactly what it "should" be. For example, $z \mapsto e^z$ is analytic and has power series $\sum_{n=0}^\infty \frac{z^n}{n!}$.
However, we would not expect your example $f_A(z)=exp(z A - \bar z A^*)$ to be analytic. To see this, let's specialize to the case where $\mathcal A=\mathbb C$ and $A=1$. Then your function is $$f(z)=exp(2i \ Im \ z),$$ where $Im \ z$ is the imaginary part of $z$. Power series should be in $z$, not in $Im \ z$ or $Re \ z$. Indeed, when we check whether the Cauchy-Riemann equations are satisfied, we see that the function $f$ is not analytic.
For details on the holomorphic functional calculus, you can take a look at Conway's A Course in Functional Analysis, Section VII.4. If you are looking for conditions equivalent to analyticity, you can find a couple in Takesaki's Theory of Operator Algebras III, google books link. (There are surely better references available, but this is what I have by my side.)