Question

[ Complex Analysis ]

Find the laurent series for 1/(1-Z2) about Z0=1.

Answer 1

Recall that for
`|z|<1`, we have

`\frac1{1-z}=\displaystyle\sum_(n\ge0)z^n`

Then

`\frac1{1-z}=-\frac1z\frac1{1-\frac1z}=-\frac1z\displaystyle\sum_(n\ge0)z^(-n)=-\sum_(n\ge0)z^(-n-1)`

valid for
`|z|>1`, so that

`\frac1{1-z^2}=-\frac1{z^2}\frac1{1-\frac1{z^2}}=-\frac1{z^2}\displaystyle\sum_(n\ge0)z^(-2n)=-\sum_(n\ge0)z^(-2n-2)`

also valid for
`|z|>1`.