What is the Laurent series of function $f(z)= 1 / (1- z ^2)$ with centre at $z=1$?

Question

What will be the Laurent series for above function?

Answer 1

Hint:

Set $z=1+u$. This function can be rewritten as

$$\frac{1}{1-z^2}=\frac 1{(1-z)(1+z)}=-\frac1{2u}\,\frac1{1+\cfrac u2}$$ and you can expand the second fraction with the geometric series.