I've been asked to find and classify all singular points (removable, pole of order n, essential, or not an isolated point). I am struggling with when infinity is a singular point and also how to recognize an essential singular point. I have provided what I think the answers to the questions I'm struggling with are for the ones I think I know. For the others, I am completely lost.
(a) $\frac{1}{(z^2+4)^2}$
I got 2i, -21 are poles of order 2 since they are zeros of 1/f of order 2
(d) $\frac{z^2+1}{e^z}$
Is z = -infinity the only pole? How do I find its type?
(e) $\frac{1}{e^z−1}$ − $\frac{1}{z}$
Do you combine them first?
(h) e$^{-z}$$cos\frac{1}{z}$
I multiplied this out to get that it = .5*(e$^i + e^-i$) so are there no singularities?
(k) sin ($\frac{1}{cos\frac{1}{z}}$) I found that cos($\frac{1}{z}$) does not have an isolatged singularity at z = 0. Does that help here? Does it have isolated singularities at z = $\frac{1}{.5(n*pi) + n*pi}$? since \frac{1}{cos\frac{1}{z}} never exceeds one, is there a singularity when cos is 1 aka at $\frac{1}{n*pi}$?
Any hints would be greatly appreciated. Thanks!