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I can get two Laurent series for that but I do not remember how to handle the annulus stuff. I did not write it down either so I will have to rediscover the method.
I don't know. The lecture notes I am looking at say "find Laurent expansions for the function f(z)" and gives the annulus ranges.
I'm not sure what you mean -- I just got f(z) into a form which I could generate the series of. How do I expand around a pole?
The principle part is the part involving negative powers. To find the Laurent you expand around one of the poles. Which pole did you use?
I know those are the poles (from the quadratic), but how can I use that here? I thought the principal part was the part of the series involving negative powers -- which part is my principal part here?
Find Laurent expansions for:
valid in the annuli;
(a) 0 ≤ |z| < 1,
(b) 1 < |z| < 3,
(c) 3 < |z|.
I've found a Laurent expansion but I'm not sure what to do about the different annulus ranges.
which forms the geometric series
which I think simplifies to
but I don't know how to address the problem of the annulus ranges given for (a), (b) and (c). I think mine is valid for the range in (a), but I don't know what to do about the others.