Hi, I'm trying to solve part b of this question. I got the laurent series centered at 0 (part a) by using the geometric series and partial fractions to manipulate the expression to match the boundary conditions, but I'm not sure how to center it at i around an annulus where we don't know the radius. How could I do this? Thank you for your help!
Hint: $\frac{1}{(z-i)(z-2)}=\frac{a}{z-i}+\frac{b}{z-2}$. Work out the constant $a$ and $b$, and notice that $\frac{a}{z-i}$ is already a Laurent series at $z=i$, so what you need to do is to figure out the expansion of $\frac{b}{z-2}$ at $z=i$.(Try $\frac{b}{(z-i)+(i-2)}$)