# Could any non-zero entire function be constantly zero on $(0,\infty)$?

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!

• Do you know the en.wikipedia.org/wiki/Identity_theorem ? – Martin R Oct 30 '17 at 21:20
• @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. – Michael Hardy Oct 30 '17 at 21:25
• @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" – Martin R Oct 30 '17 at 21:27