Question

laurent series expansion?

Answer 1

\[f(z)=\frac{ 1 }{ z(z-1)^{2} }\]

Answer 2

expanding about the point z=1,
let u=z-1 then z=u+1
\[f(z)=\frac{ 1 }{ (u+1)u ^{2} }=\frac{ 1 }{ u ^{2} }\frac{ 1 }{ 1-(-u) }\]\[f(z)=\frac{ 1 }{ u ^{2} }\sum_{n=0}^{\infty}(-1)^{n}(u)^{n}\]\[f(z)=\frac{ 1 }{ (z-1) ^{2} }\sum_{n=0}^{\infty}(-1)^{n}(z-1)^{n}\]\[f(z)=\sum_{n=0}^{\infty}\frac{ (-1)^{n} }{ (z-1)^{n-1} }\]my steps correct for z=1?

Answer 3

(z-1)^n-2 for the denominator*