(i). Determine the residue of at each of its poles
The poles arise (possibly) when:
Now I am not going to do this problem because the solutions to that equation are messy, are you sure this is the problem?
Firstly, about the parenthesis - its my first time using the inbuilt equation editor (which I have never used before) so as long as it looked ok, i was happy rather than spending ages figuring it out. please accept my apologies. I'll have a play around with it to so that i can get it right next time.
Its definitely the problem. Like you say, its quite messy but rest assured it is whats required since this question gets a total of 8% of the paper.
I am sorry, it is not messy, I first read it as a 60 instead of a 50 which made it messy.
Let me continue the poles arise possibly when:
It is in the special form where are holomorphic and .
So the residues can be computed as:
Consider the integral,
Because if we integrate this using the parametrization on we get percisely the integral you have above.
Simpify the integral,
Above we found that the poles where . Since we are working on the unit circle we see that lie outside this circle. Thus, is the only pole.
So the integral value is,
If we equal real and imaginaries we find that the real part is 0. Which is the value of the integral.