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.

  • $\begingroup$ 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

Look at the Casorati-Weierstrass Theorem.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.