I must calculate .
My attempt :
Let . I've calculated the poles of to be , , and .
I chose the curve as being the circle centered at with radius .
So I have that .
It remains to calculate the residues of , which I wasn't able to do.
I tried to express as a Laurent series but I didn't succeed in it.
I also tried to use the fact that the residues are worth but without success in evaluating the limit.
So how could I find the residues of ?
Thanks in advance... and sorry! I find all this very difficult to grasp in such a few time.
i think that it'll be easier to work if you write the roots of in exponential form:
(instead exponential you can use cis, of course)
And now the evaluation of the limits for the residues is easier, imo. For instance, the residue at we get:
(check this very carefully because there's a good chance I got it wrong)