I am asked to find all singularities for the function $f(z) = \sin(\cot(1/z))$.
Here is my attempt, would appreciate any feedback if this is correct or otherwise where I am wrong / not rigorous enough:
At $z = 0$, the function is undefined, however this is a "limit point of the poles" (or "non isolated singularity", terminology varies?) rather than an isolated singularity, since for any neighborhood of zero we can find a number of the form $1/n\pi$ such that f(z) is undefined there.
At $z = n\pi$ (for $n = \pm1, \pm2,\ldots$) we have additional singularities. Here I could use some help with reasoning. I want to say these are (isolated) essential singularities since:
- Each such z has a small neighborhood around it in which $f(z)$ is analytic (I believe the punctured disk of radius $\frac{1}{2n(n+1)}$ for example) so it is an isolated singularity;
- Since $\lim_{z \to \infty}\sin(z)$ does not exist (in particular because the real limit $\lim_{x \to \infty}\sin(x)$ does not exist) I'd like to claim the limit $\lim_{z \to \frac{1}{n\pi}}\sin(\cot(1/z))$ does not exist and therefore it is an essential singularity;
Am I on the right track? in particular for making the case for essential singularities what would be a more formal argument?