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Could any non-zero entire function $f$ be constantly zero on positive half real axis $(0,+\infty)$? I know if such $f$ is bounded over $\mathbb{C}$, Liouville's theorem says that it should be constantly zero over $\mathbb{C}$. I am just wondering what if it is not bounded over $\mathbb{C}$. Thanks!

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  • $\begingroup$ Do you know the en.wikipedia.org/wiki/Identity_theorem ? $\endgroup$ – Martin R Oct 30 '17 at 21:20
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    $\begingroup$ @MartinR : The identity theorem as stated in that Wikipedia article refers to an open set in the plane. Then interval $(0,\infty)$ is not an open set in the plane. $\endgroup$ – Michael Hardy Oct 30 '17 at 21:25
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    $\begingroup$ @MichaelHardy: And later in that article: "Specifically, if two holomorphic functions f and g on a domain D agree on a set S which has an accumulation point c in D then f = g on all of D" $\endgroup$ – Martin R Oct 30 '17 at 21:27
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Zeros of non-constant holomorphic functions are isolated, so no.

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  • $\begingroup$ Many thanks! Any reference for this? $\endgroup$ – Yanhg Oct 30 '17 at 22:22
  • $\begingroup$ @Yanhg Any textbook in complex analysis. It's a straight-forward consequence of power series representations. $\endgroup$ – mrf Oct 31 '17 at 12:30

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