Let $z_0 \in \mathbb{C}$, let $R>0$, and let $(a_m)_{m=0}^{\infty}$ be a sequence in $\mathbb{C}$ whose [power series](/page/Power%20Series) centered at $z_0$ has [radius of convergence](/theorems/262) at least $R$. Suppose that the map $f: B(z_0,R) \to \mathbb{C}$ is represented on $B(z_0,R)$ by
for every $z \in B(z_0,R)$. Then $f$ has complex derivatives of every order on $B(z_0,R)$, and, with the convention $f^{(0)}:=f$, for every integer $n \ge 0$,