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Complex Analysis Proof

  1. Apr 26, 2009 #1
    1. The problem statement, all variables and given/known data
    C = positively oriented simple closed piecewise smooth path

    Prove that:

    (1/2i)*\int_{C}\bar{z}dz

    is the area enclosed by C.

    2. Relevant equations

    *I know that the curve C is piecewise smooth so that it can be broken up into finitely many pieces so that each piece is smooth.

    *Cauchy's integral formula

    3. The attempt at a solution

    I think that I want to let some function f(t) be a continuous complex-valued function on the path C. Let g(t) be a parametrization of the curve. Frankly, I'm not sure where to go from here. I've just started doing contour integration problems and I have problems knowing where to start with proofs.

    I'm just hoping someone can point me in the right direction on where I might want to begin. Thanks.
     
  2. jcsd
  3. Apr 26, 2009 #2

    Dick

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    You can compute the area inside a contour using Green's theorem as (1/2) integral xdy-ydx. Break your complex contour into real and imaginary parts.
     
  4. Apr 26, 2009 #3

    AKG

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    A consequence of Green's Theorem is that the area enclosed by C is:

    (1/2)\int_C xdy - ydx

    which is

    (1/2)\int _0 ^1 x(t)y'(t)dt - y(t)x'(t)dt

    where g : [0,1] -> C, g(t) = x(t) + iy(t) is your parametrization of C.
     
  5. Apr 26, 2009 #4
    thank you very much for your help.
     
  6. Apr 27, 2009 #5
    Thank you again for the assistance. I have almost gotten to the conclusion that I need but I'm not sure how to proceed.

    I have shown that:

    (1/2)\int_{0}^{1} \bar{z}dz = (1/2)\int_{0}^{1} x(t)y'(t)dt-y(t)x'(t)dt + (1/2i)\int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt

    I am assuming that the integral \int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt is equal to zero because if so I will obtain the result I am looking for, namely that:

    (1/2)\int_{0}^{1} \bar{z}dz = (1/2)\int_{0}^{1} x(t)y'(t)dt-y(t)x'(t)dt

    I'm not sure how to evaluate \int_{0}^{1} x(t)x'(t)dt-y(t)y'(t)dt.

    Any ideas would be much appreciated.
     
  7. Apr 27, 2009 #6

    Dick

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    Use Green's theorem on it.
     
  8. Apr 27, 2009 #7
    Thank you sir! Solved! Not sure how to update thread title.
     
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