Example 8.2.7
Let
. We wish to expand
as a Laurent series convergent on an annulus.
We decompose
as a sum of partial fractions:
Now, we expand both partial fractions as Laurent series about 0:

This series is convergent for
.

This series is convergent for
,i.e.
.
The intersection of the two domains of convergence is empty, so we went in a wrong direction. Let us try in another
way:

This series is convergent for
, i.e.
.

This series is convergent for
, i.e.
.
The intersection of these convergence domains is the annulus displayed on Fig
4(b).
On this annulus, a Laurent series expansion of
is: