Can a non-const holomorphic function $f$ in $\mathbb C$ have no zeroes?
It seems to me that from fundamental theorem of algebra there should exist infinite number of zeroes since Taylor series (which is roughly speaking polynomial of infinite degree) converges to $f$, so the answer would be no. Am I right?

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    $\begingroup$ If $f$ is entire, so is $e^f$, which has no zeroes. $\endgroup$ – Davide Giraudo Apr 23 '13 at 19:05
  • $\begingroup$ @DavideGiraudo great counterexample. Could you maybe make this comment into an answer so it could be accepted? $\endgroup$ – Ruslan Apr 23 '13 at 19:19

There are entire functions with no zeroes, for example $z\mapsto e^z$ or more generally $z\mapsto e^{f(z)}$, where $f$ is entire.

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    $\begingroup$ To tack on to this answer: nonconstant entire functions can only omit one point in their range as per the Picard theorem. $\endgroup$ – Cameron Williams Apr 23 '13 at 19:23

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