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I have asked a question few days ago:

Entire function where at each $a \in \Bbb C$, at least one coefficient of the Taylor series at $a$ is real

The question is:

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be a holomorphic function. Suppose that for each $a\in \mathbb{C}$ at least one coefficient of the Taylor's series $f$ about $a$ is real. Then, prove $f$ being a polynomial?

Thanks for @MartinR and @Conrad for their excellent explanation.

Today, I was reviewing the proofs. I recall that the Liouville's Theorem stated that is $f$ is a bounded entire function, then $f$ is a constant.

And because being a polynomial implies that there is some $N$ such that for all $k>N$, the derivatives are $f^{(k)}(z)=0$.

I am wondering if there is a way to prove the question with using the Liouville's Theorem.

Does anyone can help me figure it out?

Zizheng Yang
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  • The above statement holds for holomorphic functions $f: D \to \Bbb C$ defined on an arbitrary (connected) domain $D \subset \Bbb D$. On the other hand, Liouville's theorem is specifically for holomorphic functions defined in the entire plane. Therefore I find it unlikely that Liouville is of use here. (It is difficult though to prove that a certain technique can not be used in a proof.) – Martin R Sep 13 '21 at 04:39
  • Interesting, is any famous papers is about proving a theorem cannot be used in a proof? Just curious. Maybe this is a good question to ask. LOL – Zizheng Yang Sep 13 '21 at 05:34
  • I have no idea ... – Martin R Sep 13 '21 at 06:38

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