# Laurent SeriesWatch

#1
Q If find the Laurent Series of f around the simple pole . What is the radius of convergence of the holomorphic part of this series?

Can anyone see if I've done this correct?

Splitting f into partial fractions: .

The 1st fraction is holomorphic in the neigbourhood of z=2. And

i.e. the geometric series.

So the Laurent Series of f around the simple pole z=2 is:

and the radius of convergence of the holomorphic part of this series (the geometric series) is R=1.
0
7 years ago
#2
The series doesn't converge around , and anyway, your Laurent series should take the form , whereas yours has the n=-1 term expanded at z=2 and the others (the 1/(1-z) bit) expanded at z=0. You need to expand about instead, to get something of the form .
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