I have the following homework problem I need help with:
Let $G=\{z\in\mathbb{C}:\, z\neq0\}$ and define $f:\, G\to G$ by $f(z)=\frac{1}{z}$.
Find all possible Laurent expansions of $f$ which are not Taylor expansions and for each such expansion specify in what set the Laurent series converges.
What I tried:
For the point $z=z_{0}$ there are Laurent expansions in $$E_{1}:=0<|z|<R$$ and in $$E_{2}:=|z|>R$$
Where $R>0$ is real.
I am unsure about in which set the Laurent series converges to $f$ , but it seems that in both cases the Laurent series of $f$ is simply $\frac{1}{z}$ and it converges in all points of $E_{1}$or $E_{2}$ (according to the case).
Am I correct ?
For $z_{0}\ne0$ the Laurent expansions which is not Taylor is in $$E_{1}:=0\leq|z|<R$$ where $R>|z_{0}|$ and $$E_{2}:=|z|>R$$ where $R<|z_{0}|$.
I am not sure about where I should of used $<$ or $\leq$ in the description of $E_{1},E_{2}$.
I am also having difficulty finding the Laurent series at $E_{i}$ and telling where it is convergent.
Can someone please help me out with the case $z\neq z_0$ and tell me if I did correctly the first case ($z=z_0$) ?