1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Jul 14, 2013 #1

    dla

    User Avatar

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Cauchy's Integral Theorem - use partial fractions to solve integral?
Loading...