<s>[INST] Laurent series for $\frac1{z^2+1}$ I have this problem:
Find the Laurent series around $z=0$, for $\dfrac{10}{(z+2)(z^2+1)}$ in the region $1<|z|<2$.
I did partial fractions and found this: $\dfrac{2}{z+2}-\dfrac{2z-4}{z^2+1}$,
then I have to know what's the Laurent series for $\dfrac{1}{z^2+1}$ to solve the problem.
Do you know that series? Thank you.
 [/INST] The answer by Santosh is not correct.

The partial fraction decomposition of $f(z)$ is
$$
f(z) = \frac{2}{z+2} - \frac{1-2i}{z+i} - \frac{1+2i}{z-i}
$$
Since we are looking for Laurent series in the annulus, $1<\lvert z\rvert < 2$, we have that $\frac{1}{\lvert z\rvert} < 1$ and $\frac{\lvert z\rvert}{2}<1$.
\begin{align}
\frac{2}{z+2} - \frac{1-2i}{z+i} - \frac{1+2i}{z-i} &=
\frac{1}{1+\frac{z}{2}} - \frac{1-2i}{z}\frac{1}{1+\frac{i}{z}} - \frac{1+2i}{z}\frac{1}{1-\frac{i}{z}}\\
&=\sum_{n=0}^{\infty}(-1)^n\biggl[\Bigl(\frac{z}{2}\Bigr)^n-\frac{1-2i}{z^{n+1}}(i)^n\biggr]-(1+2i)\sum_{n=0}^{\infty}\frac{(i)^n}{z^{n+1}}
\end{align}
</s>