I want to find the Laurent expansion(i.e. the Laurent series) of
a) in the domain
b) in the domain
I know the solution in a) is
And in b) it is
But why is that so?
I will leave the details to you but this should get you started.
Now if I factor out (z-i) this gives
Now we factor a 2i out to get
Now if we expand this in a geometric series we get
Simplifying gives
Notice that
From here just plug in and take two terms out of the series and reindex.
The 2nd one is very similar but here is a hint
Now expand this in a geometric series