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I need some help with this proposition:

If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function such that $\{z\in \mathbb{C}:f(z)=w\}$ is finite for all $w\in \mathrm{Im} (f)$ then $f$ is a polynomial.

Any hint would be appreciated.

felipeuni
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    If you want to avoid big Picard, have a look at http://math.stackexchange.com/questions/287683 – mrf Aug 28 '14 at 07:42
  • @ mrf: Unfortunately, Baire's category theorem together with Casorati-Weierstrass one are not simpler than Great Picard's theorem . – user64494 Aug 28 '14 at 08:09
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    @user64494 That depends. Casorati-Weierstrass is elementary, and if you have taken topology before complex analysis, you probably know Baire's category theorem. Anyway, the OP wanted to avoid big Picard, not necessarily replace it with something simpler. – mrf Aug 28 '14 at 08:25
  • @ mrf: You wrote "Anyway, the OP wanted to avoid big Picard, not necessarily replace it with something simpler". I don't see that in the question. This is your personal opinion. – user64494 Aug 28 '14 at 08:39
  • @mrf See his comment in the answers. – zibadawa timmy Sep 02 '14 at 23:48

2 Answers2

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Hint: If $f$ is entire then it is a polynomial if and only if it does not have an essential singularity at $\infty$.

zibadawa timmy
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Great Picard's Theorem implies that statement.

user64494
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