If \(f\) is holomorphic on a region \(\Omega\) and for each \(z_0\in\Omega\) at least one coefficient in the power series \(f(z)=\sum_{n=0}^\infty c_n(z-z_0)^n\) is zero, show that \(f\) is a polynomial.
Solution
For each \(n\ge0\), let \[ Z_n=\{z\in\Omega:f^{(n)}(z)=0\}. \] Each \(Z_n\) is closed in \(\Omega\). The hypothesis says that for every \(z_0\in\Omega\) at least one Taylor coefficient at \(z_0\) vanishes, i.e. \(f^{(n)}(z_0)=0\) for some \(n\). Hence \[ \Omega=\bigcup_{n=0}^\infty Z_n. \]
Choose a closed disk \(\overline D\Subset\Omega\). Since \(\overline D\) is a complete metric space, Baire’s theorem implies that one of the closed sets \(Z_n\cap\overline D\) has nonempty interior in \(\overline D\). Thus \(f^{(n)}\) vanishes on a nonempty open subset of \(\Omega\). By the identity theorem, \[ f^{(n)}\equiv0\quad\text{on }\Omega. \] Therefore \(f\) is a polynomial of degree at most \(n-1\) (with the case \(n=0\) giving \(f\equiv0\)).