<s>[INST] Find the Laurent series in $1<|z|<2$ ring I just started Laurant series, I have this function
$$f(z) = {z^4 + 1 \over {(z-1)(z+2)}}$$ in  $1<|z|<2$ ring, I know the theorems,but still cant figure it out, Any help with idea or transformation , welcome!
 [/INST] \begin{align}
f(z) & =z^2-z+3+\frac{2}{3(z-1)}-\frac{17}{3(z+2)} \\
& = z^2-z+3-\frac 23 \cdot \frac{1}{1-z}-\frac{17}{6} \cdot \frac{1}{1-(-z/2)} \\
& = z^2-z+3-\frac 23 \big(1+z+z^2+z^3 \cdots \big)-\frac{17}{6}\big(1-\frac 12z+\frac 14z^2-\frac18 z^3+\cdots \big) \\
& = -\frac 12-\frac 14z-\frac 38z^2+\cdots
\end{align}
</s>