This question appears really easy, I am just not sure I am undestanding it correctly.
I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?
We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)
You can do what CaptainBlank said and use the residue theorem. But here is a special case, you can rely on the Cauchy integral formula but you still need winding numbers because is not a contour (i.e. a simple closed curve).