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Complex Contour Integral

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Folks,

    How do I evaluate the integral of (z+2)/z dz for the path C= the top half of the circle |z|=2 from z=2 to z=-2.

    3. The attempt at a solution

    I take ##z=x+iy## and ##dz=dx+idy##

    Therefore ##\int_c f(z)=\int_c (1+(2/(x+iy))(dx+idy)##....not sure if I'm going the right direction

    Or do I parameterise z as ##z(t)=e^{it}##..?

    Thanks
     
  2. jcsd
  3. Apr 28, 2012 #2

    Dick

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    Parametrize as z=2e^(it). What is dz?
     
  4. Apr 29, 2012 #3
    Thank you, sorted.

    Stuck on this one. ##\int_c (x^2+ixy) dz## where C is given by ##z(t)=t^2+t^3i## for ##0\le t\le1##

    I thought of converting z to polar coordinates where ##z=r\cos \theta + ir \sin \theta## ad ##x=r\cos \theta## so we have

    ##\int_c x(x+iy)dz=\int r\cos\theta(r \cos \theta+i r\sin \theta)(-r\sin \theta d\theta+i r \cos \theta d \theta)##....the limits not sure how to approach, perhaps approach is wrong..?
     
  5. Apr 29, 2012 #4

    Dick

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    I don't think polar coordinates are any help. Just write it as an integral dt.
     
  6. Apr 29, 2012 #5
    But how do I handle the x outside the bracket when we let z=(x+iy) inside the brackets?
     
  7. Apr 29, 2012 #6

    Dick

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    If z=t^2+it^3 then x=t^2 and y=t^3, right?
     
  8. Apr 29, 2012 #7
    should have spotted that. thanks
     
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