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Math Help - Contour Integration

  1. #1
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    Contour Integration

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

    Here's my attempt.

    |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.
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    |z - 2| = i
    This is nonsense. The LHS is real. I would use residue theory or Cauchy's Integral formula on this one. What do you get when using those methods? Or what methods have you been taught so far?
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    Duplicate post.
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  4. #4
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    |z - i| = 2

    z0 = i

    (z^2 - 4)/(z^2 + 4)
    = (z^2 - 4)/((z + 2i)(z - 2i))
    = (z^2 - 4)/(z + 2i) * 1/(z - 2i)

    I'm stuck here trying to factor out 1/(z - i) from the function. I have no idea what the residue theory is. I only know Cauchy's integral formula.
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  5. #5
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    There are exactly two poles of your integrand. How many of them are inside the contour? Which ones? Your z_{0} in Cauchy's integral formula needs to be one of those poles, not necessarily the center of the circle around which you're integrating.
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  6. #6
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    I'm sorry, but my understanding of this topic is very basic. The book I'm using (Eng. Math by Kreyszig) explains the concept poorly for beginners like me.

    First of all, what exactly are poles? Are they the same thing as singularities (i.e. the surface peaks)? How do you get the appropriate z variable?

    Also, in the Wikipedia example, I don't understand how the function g is rewritten, and how the complex variable i appears out of nowhere.
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    In a rough-and-ready fashion, poles are those locations in the complex plane where your function blows up. They are places where the denominator is zero. To be able to use the Cauchy Integral Formula (CIF) here, what you need to do is compare your integrand with the integrand in the CIF, and see what the function f in the CIF should be. This should also tell you what z_{0} should be. Hint: f has to be analytic inside the contour. That means the 1/(z-a) should be the only pole of your original integrand that is inside the contour.

    Does it make more sense now?
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  8. #8
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    I understand the first part about poles, but how do we check if f is analytic inside the contour? The u and v are part of the denominator, and I don't know how to apply the Cauchy-Riemann theorem for this case.

    For my question, the poles seem to be 1/(z + 2i) and 1/(z - 2i). When you say 1/(z-a), are you referring to the original 1/(z - i)?
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  9. #9
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    Rational functions like yours are analytic everywhere there isn't a pole. You have correctly identified the poles. Which poles, if any, occur inside your contour?

    The 1/(z-a) referred to the CIF wiki page.
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  10. #10
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    i is a negative number, so |z + 2i| should be smaller than 2 since |z - i| = 2, and hence inside the contour. However, if i is treated as positive then the other pole should be inside the contour.

    I still don't understand how the g function on the Wikipedia page is rewritten because the factors don't add up.
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    i is a negative number
    No, no. i is the square root of -1. It is neither positive nor negative. Those two adjectives can only apply to real numbers, which i is not. You need to plot the circle |z-i|=2 in the complex plane, and then plot the two poles you found, and see which of them lie inside the circle.
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  12. #12
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    The circle has radius 2 with center -1 on the imaginary axis, so only 1/(z - 2i) is in the circle (assuming the fact that it's a reciprocal is ignored, because I don't know how to plot it otherwise).

    Does this mean f(z) = (z^2 - 4)/(z + 2i)? Which value do we take as the z0?
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  13. #13
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    No, no. The radius is correct, but the circle is centered at i. The rest of your reasoning, oddly enough, is more or less correct.

    Summary of technique for using Cauchy integral formula on the problem \displaystyle\oint_{C}g(z)\,dz, where there is only one pole of g(z) inside C:

    1. Determine locations of all poles of the function g(z).
    2. Find the pole that is in the contour C.
    3. Write the function g(z) as g(z)=\dfrac{f(z)}{z-z_{0}}, where f(z) is analytic (has no poles) inside the contour, and z_{0} is the pole of g(z) that is inside the contour C.
    4. Use the CIF to evaluate the integral.

    Make sense? So what do you get?
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  14. #14
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    Wow, that explains a lot.

    2pi*i*f(2i)
    = 2pi*i*(-2/i)
    = -4pi

    Thanks!
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  15. #15
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    Correct. You're welcome!
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