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Laurent Series

  1. Jan 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Say whether the indicated point is regular, an essential singularity, or a pole, and if a pole of what order it is.

    [tex]\frac{z^2-1}{(z-1)^2}, z = 1[/tex]


    2. Relevant equations



    3. The attempt at a solution
    Right now I'm just sort of stuck on how to put this into a laurent series - I can't seem to expand the denominator in a series about 1 because I keep getting infinity :(

    any hints or suggestions?
     
    Last edited: Jan 25, 2010
  2. jcsd
  3. Jan 25, 2010 #2

    Dick

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    You don't have to do a full Laurent series. Just factor the numerator and denominator and simplify.
     
  4. Jan 25, 2010 #3
    i found that it reduces to [tex]\frac{z+1}{z-1}[/tex] but again don't I have to expand this and i have the same problem?
     
  5. Jan 25, 2010 #4

    Dick

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    I don't believe it reduces to (z+1)/(z-1). Did you enter the problem wrong? If not, try that again and show you did it.
     
  6. Jan 25, 2010 #5
    oops! it should read

    [tex]\frac{z^2-1}{(z-1)^2}[/tex]
     
  7. Jan 25, 2010 #6

    Dick

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    Ok, now I believe you. Now that has the form f(z)/(z-1) where f(z)=z+1. f(1) is not zero. You can classify the singularity just using that. It's also pretty easy to show the full Laurent series if you don't believe me. Write it as (2-(1-z))/(1-z).
     
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