So, I'm trying to find the regions in which the function may be represented as a Laurent series (expanded about the origin). Then I want to find those expansions:
$$f(z)=(z^2+1)^{-1/2}$$
Now, since there's a singularity at $-i \space \space and \space \space i$, the regions must be 0<|z|<1 and 1<|z|< infinity. Well, I'm lost as to how the laurent series can be calculated. If we expand about a point that isn't the singularity, is that expansion not just the taylor series? In that case, we would form a binomial series, but I believe that only converges for 0<|z|<1. How do we obtain the expansion for 1<|z|< infinity?