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I have been thinking about the following exercise from an old complex analysis qualifier exam for some days but I still don't know how to solve it. The problem is as follows:

Show that the entire function $f(z) := z^2 + \cos{z}$ has range all of $\mathbb{C}$.

At first I thought that maybe I could use Picard's Little Theorem to get a contradiction. I thought that maybe by considering the function $e^{f(z)}$ I could get the contradiction by asssuming that $f(z)$ misses one point so that the exponential would miss two points and that would contradict Picard's Little Theorem, but since the exponential is periodic this argument doesn't work.

So my question is how can this be proved?

Thank you very much for any help.

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1 Answer 1

up vote 3 down vote accepted

I believe a similar proof as in the question What is the image near the essential singularity of z sin(1/z)? would work here too.

The function $g(z) = z + \cos (\sqrt{z})$ is of order $\frac{1}{2}$ (someone should verify that...) and so does not miss any points by Picard's first theorem (the link is a google books link).

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I will look at both links. Thank you. –  Adrián Barquero Mar 13 '12 at 0:30
@AdriánBarquero: You are welcome! –  Aryabhata Mar 13 '12 at 3:58
Does $\cos\sqrt{z}$ an entire function? –  Ylath 18 hours ago
@Ylath: Yes, I believe so. –  Aryabhata 5 hours ago

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