topology, winding numbers and fundamental groups 
Hi guys, is there another method than using winding numbers for this problem? And there are many different situations, which is pretty hard. The professor wants us to explore and play with different patterns. And is it somehow related to foundamental groups? We haven't studied that yet. Thanks for any help! Really appreciated.
 A: I assume in the rest of this answer that the question should say "closed curve". If the curve is not required to be closed, then any complex number, including the complex $\infty$, can be the result.
So with that assumption:

is there another method than using winding numbers for this problem?

Not really. You can approach it in other ways, but those essentially amount to proving the same results as winding numbers and the Cauchy Integral formula give you.

And there are many different situations, which is pretty hard.

Not really. By the full Cauchy integral formula, the value of the integral depends only on the winding number of the curve about 1 and the winding number of the curve about -1. Note though that either or both winding numbers can be negative.

And is it somehow related to foundamental groups?

Yes, but you don't need to know anything about that to solve this problem. All you need is the Cauchy Integral formula (for an arbitrary closed curve and multiple poles), partial fraction decomposition, and an understanding of what the winding number means.
