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Let $F$ be an entire function. We say that $a \in \mathbb{C} \cup \{\infty\}$ is an asymptotic value for $F$ if there exists a continuous curve going from a finite point to infinity such that $F$ tends to $a$ along that curve. Prove that for any non constant entire function $\infty$ is an asymptotic value.

I encountered this problem in an old qualifying exam. My thoughts on this are that if we assume that $\infty$ is not an asymptotic value, then for every continuous curve from a finite point to infinity $F$ tends to some finite value along that curve. One possibility is that $F$ tends to the same finite value in which case $F$ has a finite limit at $\infty$. Hence on the compact set $\mathbb{C}\cup\{\infty\}$, $F$ is bounded. Then we reach the contradiction that $F$ is bounded by Liouville's theorem. The other possibility is that $F$ does not have a limit at $\infty$ but takes on finite values in the neighborhood of $\infty$. In this case, we are probably getting a contradiction from Picard's Great Theorem.

Can someone please comment on this argument and point out possible wrong inferences or missing details?

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It is wrong. Whether $\infty$ is or is not an asymptotic value, you could have curves on which $F$ is unbounded without having a limit.

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  • $\begingroup$ I see my mistake. Can you please point to the right approach? Should I try to construct a curve on which $\infty$ is an asymptotic value? $\endgroup$ – Sourav D Aug 4 '14 at 8:18
  • $\begingroup$ Or is it more sensible to consider the behavior of any entire function at infinity. Since the function cannot have a removable singularity there, hence it will have either a pole or an essential singularity. How do we proceed from here? $\endgroup$ – Sourav D Aug 4 '14 at 8:29
  • $\begingroup$ @ Robert Israel: Could you kindly base your statement? $\endgroup$ – user64494 Aug 4 '14 at 11:52
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For example, see A. S. B. Holland, Introduction to the theory of entire functions, pp. 159-160 for the proof.

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