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!
2026-03-27 15:05:14.1774623914
Could any non-zero entire function be constantly zero on $(0,\infty)$?
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Zeros of non-constant holomorphic functions are isolated, so no.