I'm a little confused about a one part of a specific example of a Laurent series that was given by a prof. It seems like it should be pretty straightforward. I want to expand $f(z)=\frac{1}{z^2-z}$ as a Laurent series about $z=i$. I did it one way, and he did it another.
It's analytic at this point, so I think the Laurent series should just be in the form of a regular power series expansion. We know that $f(z)=\frac{1}{z-1}+\frac{1}{z}$. I agree with the example that $\frac{1}{z-1}=\frac{1}{i-1}\cdot\frac{1}{1-(-\frac{z-i}{i-1})}=\sum_{n\geq 0} \frac{(-1)^n}{(i-1)^n}(z-i)^n$. Here is where I'm confused. I would have written $\frac{1}{z}$ as: $$ \frac{1}{z}=\frac{1}{i+(z-i)}=\frac{1}{i}\cdot\frac{1}{1-(\frac{z-i}{-i})}=\sum_{n\geq0}\frac{(-1)^n}{i^{n+1}}(z-i)^n $$
No principal part, which makes sense to me.
The prof instead did: $$ \frac{1}{(z-i)+i}=\frac{1}{z-i}\cdot\frac{1}{1-\frac{-i}{z-i}}=\sum_{n\geq 0}(-i)^n(z-i)^{-(n+1)} $$
This seems to be giving us a principal part for the Laurent expansion and a non-zero residue. Can someone explain why he did this?
Thanks!
EDIT: There was a nice comment pointing out that the series I wrote and the series the prof wrote have two different domains of convergence. I'm a little confused in general then what the prof's expansion is saying about our function. Does it matter that it has an infinite principal part? What does that mean?
To expand on my original comment, let's investigate the domain of convergence of each series. First, we have
$$\frac{1}{z-1} = \sum_{n=0}^\infty (-1)^n \left(\frac{z-i}{i-1}\right)^n$$
which converges for $|z-i|<\sqrt{2}$. For $\frac{1}{z}$, we have two different candidates for our series. You have
$$\frac{1}{z} = \frac{1}{i} \sum_{n=0}^\infty (-1)^n \left(\frac{z-i}{i}\right)^n$$
which converges for $|z-i|<1$. Your professor has
$$\frac{1}{z} = \frac{1}{z-i}\sum_{n=0}^\infty (-1)^n \left(\frac{i}{z-i}\right)^n$$
which converges for $|z-i|>1$.
Putting each of these together, you have a Taylor series for $f$ which converges for $|z-i|<1$ (which can indeed, if you wish, be interpreted as a series defined on the punctured disc), whereas your professor has a Laurent series for $f$ which converges in the annulus $1<|z-i|<\sqrt{2}$. Perhaps your professor intended for you to expand your function in some annulus centered at $i$, other than the punctured disc?