laurent series expansion of terms like $\frac{1}{z}$, $\frac{1}{z^2}$

Question

I have a homework question of finding the Laurent expansion of $\frac{1}{z^2(z-1)}$ on $0<|z|<1$. I've learned to decompose the function to $\frac{A}{z}$, $\frac{B}{z^2}$, and$\frac{C}{z-1}$, and try to construct a geometry series on the last term where the ratio is less than 1. However, in the case of $\frac{A}{z}$ and $\frac{B}{z^2}$, I'm not quite sure what to do. Should I just always leave them as is (I found only one post from another forum that does it), or is it depend on which $|z|$ I'm working with? Thanks.

Answer 1

An expression like $\frac{A}{z}$ or $\frac{B}{z^2}$ (assuming $A$ and $B$ are constants) is already a Laurent series around $0$, so you don't have to change them at all! More precisely, the Laurent expansion of $\frac{A}{z}$ around $0$ is $\sum_{n\in\mathbb{Z}} c_nz^n$ where $c_{-1}=A$ and $c_n=0$ for all other $n$.