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Math Help - Evaluation of Contour integral

  1. #1
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    Evaluation of Contour integral

    Given:

    \int_C z(\overline{z}+1)^2 dz

    where C is the circle |z+1|=3 in a clockwise direction.


    I'm thinking that the answer is simple 0 since it's composite function seems to be analytic everywhere, but it's continuous nowhere, since:

    (x+iy)(x-iy+1)^2 = (x+iy)(x^2-2iyx-y^2+2x-2iy+1)  = x^3-iyx^2 +y^2x-iy^3+2x^2+2y^2+x+iy which clearly won't satisfy the Cauchy-Riemann equations
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  2. #2
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    Quote Originally Posted by lllll View Post
    Given:

    \int_C z(\overline{z}+1)^2 dz

    where C is the circle |z+1|=3 in a clockwise direction.
    You can easily do this problem directly by contour integration.
    The function g(\theta) = 1 + 3e^{i\theta} for 0\leq \theta \leq 2\pi is an image of this circle.
    We also see that \dot g (\theta) = 3ie^{i\theta}.

    Thus, the integral becomes,
    \int \limits_0^{2\pi} (1+3e^{i\theta}) (3e^{-i\theta} + 2)^2 (3ie^{i\theta}) d\theta

    Open parathesis and get your answer.
    Hint: \int \limits_{0}^{2\pi} e^{ik\theta}d\theta = 0 \text{ if }k\in \mathbb{Z}^{\times}
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