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Homework Help: Essential singularity

  1. Feb 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Hey guys.
    I need to show that this function has an essential singularity at z=0.
    I used Taylor series to get what I got, which is a series inside a series...:confused:
    And I can't see how am I suppose to show it from here.
    Any ideas guys?

    Thanks.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 13, 2009 #2

    statdad

    User Avatar
    Homework Helper

    Without seeing the result, or the original function, it's difficult to say what is right and what isn't.
     
  4. Feb 13, 2009 #3
    Sorry :smile:
     

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  5. Feb 13, 2009 #4

    Mark44

    Staff: Mentor

    Did you mean to write f(z) = cos(e^(1/2))? If so that's a constant function.

    I'm having a hard time reading your writing, as your e looks like a cross between an e and a u. Is that thing in the numerator of the exponent on e the digit 1?
     
  6. Feb 13, 2009 #5

    HallsofIvy

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    Science Advisor

    I think it is [tex]e^{1/z}[/tex] not [tex]e^{1/2}[/tex].

    Yes, write out the Taylor's series for ex and replace x with z-1 as you have done. Now, what is the definition of "essential singularity"?
     
  7. Feb 14, 2009 #6
    The Laurent series of f(x) at the point a has infinitely many negative degree terms, the thing is, how can you see that trough this series inside a series?

    Thanks.
     
  8. Feb 14, 2009 #7

    HallsofIvy

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    Science Advisor

    Oh, you have cos(e1/z). I was only looking at your first e1/z.

    Well, e1/z already has an infinite number of negative exponents. Certainly one of the coefficients will cancel out.
     
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