Calculate the Laurent development of the function in the region .

My attempt: I notice that f has 2 singularities, one in z=a, the other in z=b. By intuition, the radius of convergence of the series should be but it's not asked.

Ok so my problem is where do I center the Laurent series? There's the possibility that f is analytic in (and so the Laurent series is worth the Taylor's series) but I'm unsure since it doesn't seem bounded when and so I'd bet it's not analytic in omega.

Now with which is ok in the region omega. I could get a similar result for but it wouldn't be valid in omega so I don't really know how to go further.

So I can write and I'm stuck.