- #1
gtfitzpatrick
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Homework Statement
Calculate the laurent series expansion about he points specified, classify the singularity and sate the region of convergence for.
[itex]\frac{1}{z^2 - 1}[/itex] at (i) z=1 (ii) z=-1 (iii)z=0
Homework Equations
The Attempt at a Solution
[itex]\frac{1}{z^2 - 1} = \frac{1}{2(z-1)} - \frac{1}{2(z+1)}[/itex]
i observe that f(z) is analytic in {z≠ -1,1} so is anyalytic in -1 < [itex]\left|z\right| < 1[/itex]
@z=1 [itex]\frac{-1}{2}\frac{1}{1-(-z)} = \frac{-1}{2}\sum (-z)^n[/itex]
similarly
@z=-1 [itex]\frac{1}{2}\frac{1}{(z-1)} = \frac{-1}{2}\frac{1}{1-z)} = \frac{-1}{2}\sum (z)^n[/itex]
and
@z=0 is it just the sum of both of them? singularities are at 1, -1 and region of convergence -1 < [itex]\left|z\right| < 1[/itex]
am i anywhere near?