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Is Cauchy's integral formula applicable to this type of integral?

  1. Mar 12, 2009 #1
    1. The problem statement, all variables and given/known data

    I am trying to determine if Cauchy's integral formula will work on the following integral, where the contour C is the unit circle traversed in the counterclockwise direction.

    [tex]\oint_{C}^{}{\frac{z^2+1}{e^{iz}-1}}[/tex]


    2. Relevant equations
    See Cauchy's Integral Formula - http://en.wikipedia.org/wiki/Cauchy_integral_formula" [Broken]

    3. The attempt at a solution

    I realize that there is a pole at z=0. I realize that if I could get this integral into the form

    [tex]\frac{f(z)}{z}[/tex],

    with f(z) being analytic in and on the contour C, then I could use the formula. However, I'm not sure how to get the integrand in that form. Is it even possible to use Cauchy's integral formula on this integral, or do I need to use a different method to evaluate this integral?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Mar 12, 2009 #2

    lurflurf

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  4. Mar 12, 2009 #3
    Ok, so it appears that I need to use the residue theorem in order to evaluate this integral. I was hoping I could just use the integral formula. I haven't got to study the residue theorem yet in my text. Thanks
     
  5. Mar 12, 2009 #4

    lurflurf

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    The residue thorem is a simple use of the integral formula.
    write f(z)=[z*f(z)]/z
     
  6. Mar 13, 2009 #5
    so are these the steps:

    do a laurent expansion of denominator
    cancel with stuff in the numerator
    then the coefficient of the [itex]z^{-1}[/itex] term gives us the residue
    multiply this by [itex]2 \pi i[/itex] to give the integral's value

    im not too sure about the first of those two steps???
     
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