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Homework Help: Cauchy's Integral Theorem - use partial fractions to solve integral?

  1. Jul 14, 2013 #1


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    1. The problem statement, all variables and given/known data
    Find the integral and determine whether Cauchy's Theorem applies. Use partial fractions.

    [itex]\large \oint \frac{tan \frac{z}{2}}{z^{4} -16} dz[/itex] C the boundary of the square with vertices ±1, ±i cw

    2. Relevant equations

    3. The attempt at a solution
    I just wanted to check if approach is correct. Since tan is not analytic at ±∏ which does not lie within the square, can I just take out the tan and solve using partial then multiply it back in?

    I would have four terms since z^4-16 has four solutions but how would I evaluate each term. The integral I got is

    [itex]\huge - \oint \frac{tan \frac{z}{2}}{32(z+2)} + \oint \frac{tan \frac{z}{2}}{32(z-2)} - \oint \frac{itan \frac{z}{2}}{32(z+2i)} + \oint \frac{itan \frac{z}{2}}{32(z-2i)}[/itex]

    I can't seem to evaluate it.

    This section is before Cauchy's Integral Formula so I can't use that yet.. It only talks about Cauchy's Integral Theorem
    Last edited: Jul 14, 2013
  2. jcsd
  3. Jul 14, 2013 #2


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    Science Advisor

    All you need is the theorem "the integral of an analytic function around a closed path is 0".
    tan(z/2) is analytic for all z in the given square and the denominators are 0 only at 2, -2, 2i, and -2i. Again, those are all outside the square you are integrating over.
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