Find the Laurent series expansion of $$f=\frac {2z}{(z-1)(z-3)}$$ at the $1<|z|<3$.
Sol.: My centre is around zero.Ill use the known geometric series around zero.
$$f(z)=\frac {2z}{(z-1)(z-3)}=\frac{-1}{1-z} +\frac{3}{z-3}=\frac{1}{1-z} -\frac{1}{1-\frac{z}{3}}$$ Using $$-\sum_{n=1}^{\infty}\frac{1}{z^n}=\frac{1}{1-z}$$ for $|z| >1$ and $$\sum_{n=0}^{\infty}\frac{z^n}{3^n}=\frac{1}{1-\frac{z}{3}}$$ for $|z|<3$. Is it correct ?.If the centre was something else i would make a substitution in order to use the series around zero again?.Also fractions of the form $\frac{1}{z-centre}$ i don't need to change them since they will be able to get inside the sums?.Any other thing that needs noticing when calculating laurent series this way??