Is there some useful criterion to determine whether or not an entire function is surjective?


Maybe Picard's theorem is of help http://en.wikipedia.org/wiki/Picard_theorem

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    $\begingroup$ Indeed. And it is surjective if and only if it not of the form $e^{h(z)}+\alpha$ for a suitable constant $\alpha$ and a suitable entire function $h(z)$. $\endgroup$ Jun 3 '10 at 10:13
  • $\begingroup$ +1. And to show this, it's probably worth looking at en.wikipedia.org/wiki/Weierstrass_factorization_theorem $\endgroup$
    – dke
    Jun 3 '10 at 10:24
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    $\begingroup$ I don't see how Picard's theorem, or Roland Bacher's remark, is useful in practice to determine whether an entire function is surjective. $\endgroup$ Jun 3 '10 at 12:36
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    $\begingroup$ But certainly if there is no $\alpha \in \mathbb{C}$ such that $\frac{f'(z)}{f(z) - \alpha}$ is entire, we can conclude that $f$ is not surjective. $\endgroup$ Jun 3 '10 at 20:59
  • 2
    $\begingroup$ @Saul : We can conclude that $f$ IS surjective. $\endgroup$ Jun 4 '10 at 14:39

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