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Homework Help: Complex integral

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data
    We know sin(z) has zeros at integral multiples of pi. Let f(z) = z2/sin2(z)
    How do I find the integral of f(z) dz around C1 (C1 is the circle |z| = 1 orientated anti-clockwise) and how do I find the integral of f(z) dz around C2 (C2 is the circle |z - pi| = 1 orientated anti-clockwise).
    2. Relevant equations



    3. The attempt at a solution
    Do I use the Cauchy Integral formula for these integrals.
    If not, how would I go about doing these.
     
  2. jcsd
  3. Jun 1, 2010 #2

    lanedance

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    Homework Helper

    how about thinking residues?
     
  4. Jun 3, 2010 #3
    I got 1 as my integral.
     
  5. Jun 3, 2010 #4
    No. 1 is my value of f(z) (using L'Hopitals rule and the fact that f(z) has a removable singularity at z = 0 so this function is analytic).
    My limits of integration are 0 and 2pi so my integral is 2pi.
     
  6. Jun 3, 2010 #5
    No again. We use the residue theorem
    integral = 2 pi i (sum of the residues)
    = 2 pi i (1)
    = 2 pi i
    and for the next integral I got 4 pi^2 i
     
    Last edited: Jun 3, 2010
  7. Jun 3, 2010 #6

    Office_Shredder

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    Gold Member

    If your function is analytic how can it have a residue at 0?
     
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