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 :)
You can characterize the different kinds of singularities by limits of the function. If it has a pole in $z_0 \in \mathbb{C}$, then we have that $\lim_{z \to z_0}|f(z)| = \infty$. If it has a removable singularity, then it has a finite limit (the value which you can give the function to expand it to a holomorphic function at that point if your initial function was already holomorphic in its domain). An essential singularity is &quot;any other kind of singularity&quot;, which happens to have specific characteristics. Well, if it is just a singularity which is not removable and not a pole, then it is enough to find sequences $\{z_n\}_{n \in \mathbb{N}},\{z'_n \}_{n \in \mathbb{N}} \subset D$ (where $D$ is the domain of the given function) such that $z_n \to z_0$, $z'_n \to z_0$ with $|f(z_n)|\to \infty$ and $f(z'_n) \to a \in \mathbb{C}$.
(Actually, having an essential singularity at $z_0$ is actually equivalent to having a sequence $z_n \to z_0$ such that $lim_n f(z_n)$ does not exist.)
In this case, you can check that $z_n := \frac{i}{n}$ is such that $|f(z_n)|\to \infty$ and that $z'_n:= \frac{1}{\pi n}$ is such that $f(z'_n) \to 0$.
Hope this helps :)
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 $0.$