I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of $\cot\left(\frac{1}{z} \right)$.
You can do a really cool thing to write out the Laurent series, and from it, it looks as though $\cot\left(\frac{1}{z}\right)$ has an essential singularity at $z = 0$, but that's not even an isolated singular point is it? Because $\displaystyle \cot \left( \frac{1}{z} \right) = \displaystyle \frac{\displaystyle \cos \left( \frac{1}{z}\right) }{\displaystyle \sin \left(\frac{1}{z} \right)}$, and $\displaystyle \sin \left( \frac{1}{z}\right)$ has zeros in every neighborhood of $0$? (Not sure how to show this, though...).
As for the other singular points, how do I find them and show what type they are?
Finally,for the point at infinity, I'd want to find $\displaystyle \lim_{z \to 0}\cot\left( \frac{1}{\displaystyle \frac{1}{z}}\right) = \lim_{z \to 0} \cot z $d.n.e., since if we approach the origin along the positive real axis, the limit goes to $\infty$ and if we approach the origin along the negative real axis, the limit goes to $-\infty$, so $z=0$ is an essential singularity of $\displaystyle \cot\left( \frac{1}{\displaystyle \frac{1}{z}}\right)$, which implies that $z = \infty$ is an essential singularity of $\displaystyle \cot\left( \frac{1}{z}\right)$, right? Unless it isn't for the same reason that $z=0$ isn't an essential singularity of $\displaystyle \cot\left( \frac{1}{z}\right)$? I'm a little confused...