To help you a bit in finding the residue at z=0, since our aim is to compute the residue at the singularity z=0,
consider (write the Taylor's expansion for )
Now reduce this into a Laurent series expansion about z=0 and hence find the residue as coefficient of
I'm sorry if I was vague when I described my problem. I was on a train tapping the message on my phone.
As I stated in my first post, I actually did the expansion, but I'm not sure of how to interpret the result. When I played around with expansions I used the similarity to geometric series to get the next step from where you stopped and got this.
This is about it. I don't know what the next step is...