# Isolated singularities of this function?

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 :)

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 $0.$