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I have a function $\sin (\frac 1 z) \sin(wz)$ where $w=\exp (\frac {i\pi} {4} )$. I need to find out the singularities of this function.

My progress :-

This function will have singularities on points where it isn't defined i.e. $z=0$.

This singularity may be removable, pole, or essential.

I'm unable to see how to go further. Computations on WolframAlpha returned limit as 0 along the real axis. It returned an infinity when $z$ was assumed purely imaginary. Hence I guessed it is a essential singularity. Could someone tell me a non WolframAlpha solution/insight? Thank you :)

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Looking at the limits along the real and imaginary axes is a fine idea. That was your idea, so good for you. But why did you go to WolframAlpha to find these limits? Any student of complex analysis needs to be able to do such things on his own. OK, you "guessed" the correct answer, but it would be better to be able to prove/explain how these limits imply an essential singularity at $0.$

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  • $\begingroup$ Not an answer, should be left as a comment. $\endgroup$ – Yves Daoust Sep 2 '15 at 17:02
  • $\begingroup$ The OP asked for a non Wolfram approach. I suggested finding the limits on his own, which is a non Wolfram approach. I also told him his "guess" was correct, which is an answer, and then pointed out that there is a remaining problem: showing that the found limits imply an essential singularity. (You MSE vigilantes ... geesh.) $\endgroup$ – zhw. Sep 2 '15 at 17:09
  • $\begingroup$ could you help me find these limits? $\endgroup$ – martianwars Sep 2 '15 at 17:18
  • $\begingroup$ If you try to understand complex analysis with WolframAlpha, you will not go very far. Ever heard about Laurent series? $\endgroup$ – Karl Sep 2 '15 at 17:33
  • $\begingroup$ yeah I have. How will that help find these limits? How do I find this Laurent Series in any case? $\endgroup$ – martianwars Sep 2 '15 at 18:01

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