Determining Region of Convergence for the Laurent Series of $f(z) = \frac{1}{(z-1)(z^2 + 1)}$

Question

Determine the region of convergence for the Laurent series $$\frac{1}{(z-1)(z^2 + 1)} = \sum_{n = -\infty}^\infty a_n (z-1)^n$$

I know that the series can not converge at $z = 1$ and $z = \pm i$, as those are singularities of the function. However I'm not sure how to determine the region of convergence more precisely than that.

Any help is appreciated.

Answer 1

If you mark the three singularities (simple poles, in this case) on the complex plane, what is the radius of the largest circle centred at $z=1$ within which the function is analytic (excluding $z=1$, of course)? That will give you the inner region of convergence. The outer region (where the series expansion is different, I.e. has different coefficients $a_n$) is then straightforwardly identified.