The exercise: Find the laurent expansion of $f(z)=\frac{z}{z^2+1}$ in $K_{1,2}(-i)$.
My thoughts: $K_{1,2}(-i)$ denotes the annulus. 1 and 2 the radiuses. First thing I did is decompose in partial fractions. I.e. $f(z)=\frac{z}{z^2+1}=\frac{\frac{1}{2}}{z+i}+\frac{\frac{1}{2}}{z-i}$ Usually I rewrite $K_{1,2}(-i)$ as $\{z\in\mathbb{C}:0<|z+i|<\sqrt3 = |\sqrt2 +i|\}$. And then I rewrite the denominator in terms of $|z+i|$ like this: $\frac{1/2}{z-i}=\frac{1/2}{(z+i)-(2i)}=\ldots$ but this time it somehow doesn't work. Especially with $\frac{1/2}{z+i}$. I think I'am missing something crucial.