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Complex Variables: Poles

  1. Sep 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine the nature of the singularity at z=0


    2. Relevant equations
    [itex] f(z)=\frac{1}{cos(z)}+\frac{1}{z} [/itex]



    3. The attempt at a solution
    by expanding into series:

    [itex] f(z)=\Sigma_{n=0}^{\infty} \frac{(2n)! (-1)^n}{x^{2n}} + \Sigma_{n=0}^{\infty} (-1)^n (z-1)^n[/itex]
    Now [itex] \frac{1}{z}[/itex] has no principle part, [tex] b_m=0[/itex].
    This leaves the only principle part from cos. [itex] b_m=(2m)! (-1)^m[/itex]. There are infinite bm
    so the behaviour is an essential singularity.

    I don't feel too confident about this answer. I feel I have overlooked a step.
     
  2. jcsd
  3. Sep 9, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    Ouch. If you are looking for a singularity at z=0, why are you expanding 1/z around z=1? And cos(0)=1, it's not a singularity of 1/cos(z) at all. And you can't invert a power series by inverting each term in the power series. 1/(a+b) is not equal to 1/a+1/b.
     
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