How to integrate $ze^{-z}$ over the unit circle?

Question

How can I integrate $ze^{-z}$ over the unit circle? I have tried expanding $z$ using $e^{i\theta}$ as well as $\cos(\theta)+i\sin(\theta)$. But I keep ending up with very long equations to calculate. I feel I am missing an important fact in calculating this but am not sure what?

Answer 1

Your best friend here is Cauchy's integral theorem, stating that the integral around a loop is zero as long as the function is analytic. See the linked page. In more generality, you can also apply the residue theorem, but it's not necessary here.