= Solution
The complex <Taylor series> theorem states that if $f$ is <holomorphic> in $|z-z_0|<R$, then
$$
\boxed{f(z)=\sum_{n=0}^{\infty}\frac{f^{(n)}(z_0)}{n!}(z-z_0)^n\quad (|z-z_0|<R).}
$$
The <power series> converges absolutely there and uniformly on each smaller closed disc; its coefficients are unique. For $|z-z_0|<r<R$, the <Cauchy integral formula> also gives $f^{(n)}(z_0)/n!=(2\pi i)^{-1}\int_{|\zeta-z_0|=r}f(\zeta)(\zeta-z_0)^{-n-1}\,d\zeta$. This is the analytic, rather than finite real-variable, form of <Taylor theorem> relevant here.
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