Computing the Laurent series for $1/z^2$ about $z_0=1$

Question

I'm trying to compute the Laurent series of $f(z) = 1/z^2$ about the point $z_0=1$. Looking at my notes, it appears that I need to compute a series for $|z-1| < 1$ and one for $|z-1|>1$, due to the singularity at the point $1$.

Could someone show me how to compute each of these series? I'm a bit confused on where to begin.

EDIT: The problem says to "Write all Laurent series of the following functions on annuli centered at $z_0$," so I feel like there should be two series: one valid for $|z-1|<1$ and another valid for $|z-1|>1$.

Answer 1

$z_0=1$ is not a singularity for $f(z)=\frac{1}{z^2}$, hence the Laurent series is just a Taylor series, and since: $$\frac{1}{(1-z)^2} = 1+2z+3z^2+\ldots = \sum_{j=0}^{+\infty}(j+1)z^j$$ around the origin, you have: $$\frac{1}{z^2}=\sum_{j=0}^{+\infty}(-1)^j(j+1)(z-1)^j$$ around one. You can check that the radius of convergence, one, is exactly the distance from the closest singularity, the origin.

Answer 2

The result you are after can be found using the same method as in the answer above.

For $|z| > 1$ we have

$$\frac{1}{(1-z)^2} = \frac{1}{z^2}\frac{1}{(1-1/z)^2} = \sum_{j=0}^\infty (j+1)\left(\frac{1}{z}\right)^{j+2}$$

so

$$\frac{1}{z^2} = \sum_{j=0}^\infty (-1)^j(j+1)\left(\frac{1}{z-1}\right)^{j+2}$$

for $|z-1| > 1$