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Contour Integration

  1. Oct 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Integrate (z^2 - 4)/(z^2 + 4) counterclockwise around the circle |z - i| = 2.

    2. Relevant equations

    Cauchy's integral formula

    3. The attempt at a solution

    |z - i| = 2
    |z - 2| = i

    z0 = 2

    (z^2 - 4)/(z^2 + 4)
    = ((z + 2)(z - 2))/(z^2 + 4)
    = (z + 2)/(z^2 + 4) * i

    f(z) = (z + 2)/(z^2 + 4)

    2i*pi*f(z0) = 2i*pi*(1/2) = i*pi

    The answer is -4*pi. Please tell me what I'm doing wrong.
     
  2. jcsd
  3. Oct 27, 2010 #2
    I can give you a very good piece of advice: you're got to make that look nicer so that it's easier to "see":

    [tex]\mathop\oint\limits_{|z-i|=2} \frac{z^2-4}{z^2+4}dz=\mathop\oint\limits_{|z-i|=2} \frac{z^2-4}{(z+2i)(z-2i)}dz=\mathop\oint\limits_{|z-i|=2} \frac{z^2-4}{z+2i}\frac{1}{z-2i}dz[/tex]

    Now we in the big house. Can you now just apply Cauchy's Integral formula with:

    [tex]f(z)=\frac{z^2-4}{z+2i}[/tex]
     
  4. Oct 27, 2010 #3
    But isn't the factor z - i? How can you factor z - 2i?

    Also, 2*pi*i*f(i) = 2*pi*i*(-5/3i) = -10*pi/3
     
    Last edited: Oct 28, 2010
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