complex analysis - Compute $int_Gamma frac{e^frac{1}{z}}{z-1}dz$, where $Gamma$ is the circle $|z-1|lefrac{3}{2}$, positively oriented.

Compute $\int_\Gamma \frac{e^\frac{1}{z}}{z-1}dz$, where $\Gamma$ is the circle $|z-1|\le\frac{3}{2}$, positively oriented.

The numerator is not analytic in $\Gamma$ so we can't use Cauchy integral formula. I'm thinking maybe I shold use residue theorem. But then I have these two questions:

1. Should I look for the Laurent series of $e^\frac{1}{z}$ around $z=0$? What if I look for the series around some other points in $\Gamma$?


2. After I find the Laurent series of $e^\frac{1}{z}$, how do I find the Laurent series for the $\frac{e^\frac{1}{z}}{z-1}$?

Answer

The residue in the essential singularity can be computed trough Taylor's expansions:

$$e^{\frac{1}{z}}=1+\frac{1}{z}+\frac{1}{2z^2}+\ldots,$$
$$\frac{1}{z-1}=-1-z-z^2-z^3-\ldots,$$
from which (just multiply the two series):
$$\operatorname{Res}\left(\frac{e^{\frac{1}{z}}}{z-1},z=0\right)=-\sum_{k\geq 1}\frac{1}{k!}=\color{red}{1-e}.$$