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Math Help - Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

  1. #1
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    Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

    Let Arg(w) denote that value of the argument between -π and π (inclusive). Show
    that:

    Arg[(z-1)/(z+1)] = { π/2, if Im(z) > 0 or -π/2 ,if Im(z) < 0.

    where z is a point on the unit circle ∣z= 1.

    My attempt: I know
    Arg[(z-1)/(z+1)] = Arg(z-1) - Arg(z+1) and the arg(w) = arctan(b/a), but im stuck.

    Please help

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  2. #2
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    Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

    Quote Originally Posted by habsfan31 View Post
    Let Arg(w) denote that value of the argument between -π and π (inclusive). Show
    that: Arg[(z-1)/(z+1)] = { π/2, if Im(z) > 0 or -π/2 ,if Im(z) < 0.
    where z is a point on the unit circle ∣z[/FONT]
    Know that \frac{z-1}{z+1}=\frac{(z-1)(\overline{z}+1)}{|z+1|^2}=\frac{z\overline{z}+z-\overline{z}-1}{|z+1|^2}=\frac{2\text{Im}(z)i}{|z+1|^2}
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  3. #3
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    Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

    Quote Originally Posted by Plato View Post
    Know that \frac{z-1}{z+1}=\frac{(z-1)(\overline{z}+1)}{|z+1|^2}=\frac{z\overline{z}+z-\overline{z}-1}{|z+1|^2}=\frac{2\text{Im}(z)i}{|z+1|^2}
    I understand what you did, but not sure why you did it and what im supposed to do next.
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  4. #4
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    Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

    Quote Originally Posted by habsfan31 View Post
    but not sure why you did it and what im supposed to do next.
    I did it to answer the question: \text{Arg}\left(\frac{z-1}{z+1}\right)=~?
    Do you understand that \text{Arg}(z) is a real number?
    Do you understand that -\pi<\text{Arg}(z)\le\pi~?
    Do you understand that if r\in\mathbb{R} then \text{Arg}(ri)=\pm\frac{\pi}{2}~?

    Because \frac{z-1}{z+1}=\frac{2\text{Im}(z)i}{|z+1|^2} and \text{Im}\left(\frac{2\text{Im}(z)i}{|z+1|^2} \right)=\frac{2\text{Im}(z)}{|z+1|^2}\in\mathbb{R}
    then \text{Arg}\left(\frac{z-1}{z+1}\right)=\pm\frac{\pi}{2}
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  5. #5
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    Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

    Got it thanks!
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