## When are entire functions surjective?

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

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## 1 Answer

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

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6
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 $h(z)$. – Roland Bacher Jun 3 2010 at 10:13
+1. And to show this, it's probably worth looking at en.wikipedia.org/wiki/… – dke Jun 3 2010 at 10:24

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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. – Pete L. Clark Jun 3 2010 at 12:36
@Pete L. Clark: Hence the 'maybe' in the post. I was thinking about a useful/practical criterion but nothing came to mind. – babubba Jun 3 2010 at 12:51
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. – Saul Glasman Jun 3 2010 at 20:59
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