Lemniscate Figure Eight Question 
This is a problem on a previous examination.
My question is on part C in which I have no clue how to answer. Any hints? Thank you. 
 A: Very nice exercise! 
There is deeper reasons why the answer of C is NO, but is this context, you have to use A and B:
in A, you compute the residue of f at 1 and -1, then take the sum cause the index is 1 for each,
in B, you take the difference cause the index is 1 at 1 and -1 at -1,
so you find 2 differents results.
You conclude with the fact that the integral is homotopy (here continuous deformation) invariant when the two singularities are contained in the compact complement of the closed curve you are integrating on! You knew this fact? Or we can prove it, but it's probably in your complex analysis lecture.
A: Suppose you had such a deformation $\gamma_s$ as $s$ goes from $0$ to $1$, with $\gamma_0 = \gamma$ and $\gamma_1 = \mu$. You could compute the integral over each $\gamma_s$, and it would vary continuously as a function of $s$. It would also be constant as a function of $s$ (each one can be computed by the residue theorem). But parts a and b show that the integral for $\gamma_0$ and $\gamma_1$ are different. So there cannot be a deformation. 
