integrals of the function using residue theorem

Question

how to find the integrals of $z^n exp(1/z)$(where n is any integer), around the circle $|z|=1$ traversed once in the positive sense. I used the residue theorem and the answer I get is $2\pi i ln(2)$. But I always get this kind of problem wrong... If you get a different answer, could you show the procedure? Thank you very much.

Answer 1

Hint: Use the Laurent series: $\exp(1/z)=\sum_{n\geq 0} \frac{1}{n!} \frac{1}{z^n}$

Answer 2

For $z\in D(0,1)^*$, one has: $$z^n\exp(1/z)=z^n\sum_{k=0}^{\infty}\frac{1}{k!z^k}.$$ Hence, examining the term of order $-1$ in the above developpment, one gets: $$\textrm{Res}(z^n\exp(1/z),z=0)=\frac{1}{(n+1)!}$$