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I guess I need to show that $z_0$ cannot be a removable singularity or an essential singularity.

For the case of removable singularity, can I prove it by this method: Since $z_0$ is removable, when $f(x)$ is expanded into Laurent expansion, only the analytic part ($k \ge 0$ ones) remains, so $f(x)$ cannot go to infinity as $z$ goes to $z_0$.

And I have no idea on how to prove $z_0$ cannot be an essential singularity.

Thanks for the help.

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  • $\begingroup$ this is done in any book about complex analysis, e.g. rudin's Real and complex analysis $\endgroup$ – Glougloubarbaki May 26 '13 at 17:11
  • $\begingroup$ Take a look at Big Picard for the second part. It will give you a clear idea of how to approach it. $\endgroup$ – Cameron Williams May 26 '13 at 18:01
  • $\begingroup$ Is f given to be holomorphic in some neighbourhood around $z_{0}$? $\endgroup$ – Arkya Chatterjee Aug 31 '16 at 18:06
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Look at the Casorati-Weierstrass Theorem.

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