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
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