If You want to follow a very confortable way, then...
(1)
It is easy to verify that (1) converges for ...
Kind regards
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.
I was actually looking for help in the annulus , but I believe I found out how to do it. (Found a pretty good video on YouTube explaining Laurent series--)
stays as it is since is the center of the "annulus" (punctured disk). (The 'why' I haven't completely figured out yet.)
For and we already know , so that gives the geometric series .
So we have . Hurray, case closed!