= Solution
The complex <Taylor theorem> says that an <analytic function> has the expansion
$$
f(z)=\sum_{m=0}^{\infty}\frac{f^{(m)}(z_0)}{m!}(z-z_0)^m
$$
throughout every disk centred at $z_0$ contained in $D$. In particular it converges for $|z-z_0|<\operatorname{dist}(z_0,\mathbb C\setminus D)$, and uniformly on every smaller closed disk. The coefficients are uniquely its <derivatives> divided by factorials. The convergence radius can be larger if the function extends beyond the boundary of $D$; the theorem does not say that the given domain determines its exact radius.
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