Hello. My book doesn't explain Laurent series expansions very well and so I was hoping for some help figuring out a problem from the exercises. The answers are in the book so I know I got the first two - I just don't understand the last two.

'Find a Laurent series for the function in each of the following domains:

a. ...'

For this one I did a partial fraction decomp and got .

Finding a geometric series for the second term ... and since , it has a geometric series representation .

So, (So because z=0 is not defined in the annulus, 1/z is analytic - so it's OK to use and also has no series representation?) .

That and the domain (b) I got OK - for (b) I factored out from to get a geometric series.

My problem is when you move the annulus center to the other singularity...

c. { = } and d. --

For c. I assume you can still use with no problems since the annulus doesn't includ z=0, but I am having a hard time trying to find a geometric series for .

Reverse triangle inequality didn't get me (I got - although I have doubts about that being correct); I don't think I can simply factor out to get because it's possible for which would make ... so I just don't know what to try next.