I must find the Laurent series of the function for 3 different regions: , and .

My attempt: I've found out that g has 3 singularities (at and ) and is I think otherwise analytic.

So for the first region, there is 1 singularity inside it (just at the center of the region).

So I think I should write the Laurent series centered at the singularity .

I've tried many things like rewriting but I realized it was better to keep it as it was. I was thinking about writing as an infinite series (now I realize that as it should be centered in i, I should have factored out by (z-i)...) and then divide by z the whole series. I'm going into circles and I don't click yet on how to get it. I'd like a tip.