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Complex analysis integration

  1. Dec 1, 2011 #1
    compute the integral ∫Cr (z - z0)n dz,
    with an integer and Cr the circle │z - z0│= r traversed once in the counterclockwise direction

    Solution:

    A suitable parametrization for Cr is give by z(t)= z0 + reit 0≤t≤2π
    ........
    .......

    My question is , how to find that suitable z(t)?
    i have no idea how to find out the z(t), it just pop out in the solution.

    Thanks
     
  2. jcsd
  3. Dec 1, 2011 #2
    It's a circle in the real-imaginary plane.
    A circle in the xy-plane can be parametrized by
    x=r*cos(t)
    y=r*sin(t)
    with 0<=t<=2pi
    This comes down to the fact that cos^2(t)+sin^(t)=1 as a circle is defined by x^2+y^2=R^2.
    Now bear in mind Euler's formula
    e^{ix}=cos(x)+i sin(x)
    Alright?
     
  4. Dec 1, 2011 #3
    so if C is the circle of │z - 2i │= 4
    z(t) = 2i + 4e^it ?
    i am doing it right?
     
  5. Dec 1, 2011 #4
    You got it ;)
     
  6. Dec 1, 2011 #5
    thanks!!!!
     
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