Find and classify the singularities of the following functions in $\mathbb{C}$:
$\frac{1}{z(e^{\frac{1}{z}}+1)}$
$\frac{1}{(z^2+1)(z-1)^2}-\frac{1}{4(z-i)}$
OK, so I think the first is the easier (perhaps). There's clearly an essential singularity at the origin caused by the exponential. However, I think there are also singularities where $e^{\frac{1}{z}}=-1$, which occurs when $z=\frac{1}{(2n+1) \pi}$ for $n \in \mathbb{Z}$, though I am not sure how to classify there. Help with that would be very appreciated.
For the second, we can split it into $\frac{1}{4(z+i)}-\frac{1}{2(z-1)}+\frac{1}{2(z-1)^2}$, which makes the position of the poles clear; at $-i, 1$. Is it the case that the pole at $-i$ is simple, and the pole at $1$ is a double pole. That seems to be the case.
Any help/verification would be very helpful. Thanks in advance.