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Math Help - Complex Variables

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    Complex Variables

    Compute the following line integral:

    γ(z+3z+4)dz, where γ is the circle |z|=2 oriented counter-clockwise.

    --Taking a Complex Variables course and I am completely lost, and there is no solution manual, or answers at the back of the book for even #s!
    Last edited by zhart; June 12th 2013 at 03:57 PM.
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    Re: Complex Variables

    Quote Originally Posted by zhart View Post
    Compute the following line integral:

    γ(z+3z+4)dz, where γ is the circle |z|=2 oriented counter-clockwise.

    --Taking a Complex Variables course and I am completely lost, and there is no solution manual, or answers at the back of the book for even #s!
    Give the following substitution a try: z = 2e^{i \theta}

    -Dan
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  3. #3
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    Re: Complex Variables

    I presume that you know that a complex number can be written in the form re^{-i\theta} where r is |z|, the distance from the origin ( 0) to the given point and \theta is the "argument", that angle the line from the origin to z makes with the positive real axis (positive x axis). If z= a+ ib, the r= \sqrt{a^2+ b^2}, \theta= arctan(b/a).

    The point is that any point on "the circle |z|= 2" can be written z= 2e^{i\theta} so z^2+ 3z+ 4= 4e^{2i\theta}+ 6e^{I\theta}+ 4. dz= 2ie^{I\theta}d\theta. The "oriented counter-clockwise" means the integral is from 0 to 2\pi.
    Last edited by HallsofIvy; June 12th 2013 at 04:29 PM.
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  4. #4
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    Re: Complex Variables

    Quote Originally Posted by zhart View Post
    Compute the following line integral:

    γ(z+3z+4)dz, where γ is the circle |z|=2 oriented counter-clockwise.

    --Taking a Complex Variables course and I am completely lost, and there is no solution manual, or answers at the back of the book for even #s!
    It's been a while since I did Complex Analysis, but since the contour is closed and the function doesn't have any singular points in it, doesn't the integral have to equal 0 by Cauchy's Theorem?
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    Re: Complex Variables

    Quote Originally Posted by Prove It View Post
    It's been a while since I did Complex Analysis, but since the contour is closed and the function doesn't have any singular points in it, doesn't the integral have to equal 0 by Cauchy's Theorem?
    I agree. It is zero.
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