I have the function $\frac{z+2}{z(z+1)}$ which I am trying to find the integral for a circle of radius 3 centered at the origin. I'm able to use Cauchy's Second Residue theorem to find it to be $2 \pi i$, but let's say I want to find it by just expanding the function into an appropriate Laurent Series and finding the residue through there. How would you go about expanding this into a Laurent series centered around $z=-1$? I can only find the residue for the simple pole $z=0$.
2026-04-02 10:45:31.1775126731
Finding Residue at $z=-1$ using Laurent Series
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First, consider $\frac{z+2}z=1+\frac2z$. If $|z+1|<1$, then\begin{align}1+\frac2z&=1-\frac2{-z}\\&=1-\frac2{1-(z+1)}\\&=1-2\sum_{n=0}^\infty(z+1)^n\\&=-1-2\sum_{n=1}^\infty(z+1)^n\end{align}and therefore$$\frac{z+2}{z(z+1)}=\frac{-1}{z+1}-2-2(z+1)+\cdots$$So, the residue at $-1$ is $-1$.