Please help me with this one guys, I am stuck like a truck trying to get out of thick mud.
Evaluate:
$\int_{\gamma } z^{n}e^{\frac{1}{z}}dz$
$\gamma$ is the circle f radius 1 centered at 0 and traveled once in the counterclockwise direction.
obviously in the complex plane
This is what I've got, but I don't know if it is correct.
$$\int\limits_\gamma z^n e^{\tfrac{1}{z}}dz=-\int\limits_{\gamma^{-}} w^{-n} e^{w}\dfrac{dw}{w^2}=\int\limits_{\gamma^{+}} \dfrac{e^{w}}{w^{n+2}}dw=2\pi i \underset{w=0}{\operatorname{res}}{\dfrac{e^{w}}{w^{n+2}}}.$$
and also:
$$z^n e^{1/z} = z^n \sum_{k=0}^{\infty} \dfrac1{k! \cdot z^k} = \sum_{k=0}^{\infty} \dfrac{z^{n-k}}{k!} = \sum_{k=0}^{n} \dfrac{z^{n-k}}{k!} + \dfrac1{(n+1)! \cdot z} + \sum_{k=n+2}^{\infty} \dfrac{z^{n-k}}{k!}$$