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Interesting complex variables problem

  1. Feb 27, 2008 #1

    nicksauce

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    1. The problem statement, all variables and given/known data
    Let [tex]\Omega[/tex] be a bounded domain in C whose boundary is a curve z = z(t), a<=t<=b, and let [tex]A(\Omega)[/tex] be the area of [tex]\Omega[/tex]. Prove that

    [tex]A(\Omega) = \frac{1}{2}\int^b_a |z(t)|^2 Im(\frac{z'(t)}{z(t)})dt[/tex]


    2. Relevant equations



    3. The attempt at a solution
    Not even sure where to start on this. Any tips?
     
  2. jcsd
  3. Feb 27, 2008 #2

    Dick

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    Yes. [tex]A(\Omega) = \frac{1}{2}\int_C x dy-y dx[/tex]. That's a well known expression for calculating area using Green's theorem. If you express z(t)=x(t)+iy(t), that's what your expression reduces to.
     
  4. Feb 27, 2008 #3

    nicksauce

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    Ahh right, thank you
     
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