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Laurent Expansion of sin(1-1/z)

  • Thread starter nicksauce
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nicksauce
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1. Homework Statement
Find the Laurent expansion of [tex]f(z) = \sin(1-\frac{1}{z})[/tex] about z = 0, and state the annulus of convergence.


2. Homework Equations



3. The Attempt at a Solution
I tried doing the regular expansion of sin(z), then applying the binomial expansion on the (1-1/z)^n terms, but I can't help but feel that there's a better way to approach the problem. Any thoughts?
 

Answers and Replies

HallsofIvy
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I would suggest that you use the sine sum formula to write it as
[tex]sin(1-\frac{1}{z})= sin(1)cos(\frac{1}{z})- cos(1)sin(\frac{1}{z})[/tex]
and then use the Taylor's series for sine and cosin.
 
nicksauce
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Boy don't I feel dumb. Thanks!
 
is this sentence correct?
sin(1/z)=1/z-1/z^3*3!+1/z^5*5!+-......
for laurent series?
 

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