use this give you the exact expansion i think is the best posible
Get the Laurent series expansion or f(z) = in the intervals;
(i) |z-1|<1 and
(ii) 1<|z-1|< 2
Now I don't have much trouble finding the Laurent series expansions but am not sure how to do it for specific intervals.
For example in (ii) of this problem i would get the Laurent expansions for
for interval 1<|z-1|<2
and then get the expansions for
for 1<|z-1|<2,and add them all together. The textbook(correctly, just not sure why yet) just uses the expansion of for |z-1|<2 and adds to it the expansion of
for 1<|z-1|. I'm not sure as to the procedure or the thinking behind getting Laurent expansions for these types of intervals
r1<|z-zo|< r2. Any help is much appreciated.
Now we take into account that is...
... and that for is...
Deriving (3) and inserting the result in (1) we obtain...
... that is the result You are searching for. The (4) converges for and setting You obtain the Laurent expansion of the original f(*) in z...