Here is a problem in my complex analysis notes, i.e. compute the residue of $\frac{1}{e^{\frac{1}{z}}-1}$ around $z=0$. I think that this isolated singularity is an essential singularity. The problem is to find the Laurent expansion of it near $z=0$. But the difficulty here is that I can't use those formulas such as $\frac{1}{1-x}=1+x+x^2+...$ since here the variable doesn't tend to zero.
Can anyone give me some suggestions! Thanks in advance!
That cannot be done, since $0$ is not an isolated singularity of your function. This follows from the fact that, if $f(z)=\dfrac1{e^{1/z}-1}$, then every point of the form $\dfrac1{2n\pi i}$ ($n\in\Bbb Z\setminus\{0\}$) is an isolated singularity of $f$.