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I have an entire function which is not a polynomial. Is there a way to use the Casorati-Weierstrass theorem to prove there exists a point $z_0$ such that every coefficient of the Taylor series at $z_0$ is not zero?

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up vote 5 down vote accepted

The set of points where the $n$-th Taylor coefficient is zero is the set $D_n = \{w \in \mathbb{C}\,:\,f^{(n)}(w) = 0\}$. This is a closed discrete set, hence it is countable (because if it were not discrete the identity theorem would imply that $f^{(n)} \equiv 0$, hence $f$ would be a polynomial of degree at most $n-1$). Thus, the set of points where at least one Taylor coefficient is zero is the countable set $D = \bigcup_{n=0}^\infty D_n$. Since $\mathbb{C}$ is uncountable, $\mathbb{C} \smallsetminus D$ is non-empty.

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At the moment I don't see an immediate way to answer your question using the Casorati-Weierstrass theorem. – t.b. Apr 15 '12 at 23:36

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