I am having difficulties finding Laurent series of the above function, around these two domains: $$0<|z-1|<1$$ and $$|z-1|>1$$
The function $f(z)$ takes the form $\dfrac{1}{z}-\dfrac{1}{z-1}$.
And I know that I can find the Laurent series of the following general function around $z_0=0$ when my domain includes $z_0=0$:
$$\dfrac{1}{z-a}=\dfrac{1}{z}\cdot \dfrac{1}{1-\frac{a}{z}}=\sum_{n=1}^\infty\frac{a^{n-1}}{z^n}$$
But now I should evaluate the expansion around $z_0=1$. How can this be done?
Should I expect a Laurent series for the left partial fraction term?
An explanation about this would be very respected. It just that I didn't find a similar question when searching for it.