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While trying assignments of complex analysis I am unable to solve this particular question.

Does there exists a non-constant bounded analytic function on $\mathbb{C} $/{0} ?

As function is not entire so lioville theorem can't be applied . So I think there might exist a function but I am unable to find any.

Kindly help.

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  • $\begingroup$ Do you mean set exclusion? Then first apply Riemann’s extension theorem. $\endgroup$
    – WimC
    Jul 4, 2020 at 16:36

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There is not. Any such function $f$ would be bounded near $0$. So, by Riemann's extension theorem, $f$ can be extended to an analytical function $\hat{f}$ in $\Bbb{C}$. But $\hat{f}$ is bounded and entire, so it is constant. Since $f=\hat{f}$ in $\Bbb{C} \setminus \{0\}$, $f$ is constant too.

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  • $\begingroup$ Augusto can you please tell where can I study Riemann extension theorem ? It is not part of our course. $\endgroup$
    – user775699
    Jul 6, 2020 at 17:23
  • $\begingroup$ Take a look at en.wikipedia.org/wiki/… $\endgroup$ Jul 6, 2020 at 20:02

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