How could I calculate the integration $I=\oint_{|z|=2} \frac{e^{z}}{z^{2}|z-1|^{2}} d z $? How could I calculate the integration? I’ve tried Cauchy’s Integral Formula, but how can it work with the $|z-1|$

$|z-1|^2 = (z-1)(\bar z - 1) = (z-1)(4/z-1)$ on the circle $|z|=2\Leftrightarrow z\bar z = 4$. Now you can apply the integral formula or residue theorem.
