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Complex analysis: Integration About Singularity

  1. Sep 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate the integral ∫sin(z)/(z^2-4) dz about the contour C such that it is a circle of radius 2 centered at z = 2.

    2. Relevant equations

    All theorems of complex analysis except residue theorem.

    3. The attempt at a solution

    There is a singularity at z = 2, so we avoid it by drawing a bridge going from the outer contour now renamed C1, to another circular contour C2 with opposite orientation going around z = 2. The bridges have opposite orientation but same direction, so they cancel out, and the distance between them can be made arbitrarily small.

    Therefore the summation of these 2 integrals is C, within which the function is analytic.

    Then we recognize that ∫ C1 + ∫ C2 = ∫C = 0.

    However I don't know how to apply this directly to solve the problem.

    I tried a different way. Lets take the analytic contour C.

    Note that f(z) = sin(z)/(z^2-4) = sin(z)/{ (z-2)(z+2) } = sin(z)(z+2)^-1/(z-2)

    ∫f(z)dz is now in the form ∫f(z)/(z-a) dz = 2πi f(a). Here a = 2, f(z) is now sin(z)(z+2)^-1

    Substitute 2 into the equation and we have the integral around C is equal to
    2πi*sin(2)/4 = πi sin(2)/2.

    However this is for the entire contour C, while what I'm looking for is C1, the outer contour.

    Can I just say that since C2 can be made arbitrarily small, the integral around it must be zero, and the answer I got for C is the final answer? If not what should my next step be?
     
  2. jcsd
  3. Sep 23, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    You have the Cauchy Integral Formula, yes? If so then just split 1/(x^2-4) using partial fractions and go from there.
     
  4. Sep 23, 2012 #3
    thank you greatly. I already split it up with the Cauchy integral formula and got the answer. I was just wondering if that's my final answer. I think it is now.
     
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