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Math Help - Complex Analysis - Points of Discontinuity

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    Complex Analysis - Points of Discontinuity

    Find all points of discontinuity of the function g(z)=Arg(z^2).

    I have no idea what to do with this problem. Help!
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    Re: Complex Analysis - Points of Discontinuity

    Quote Originally Posted by tarheelborn View Post
    Find all points of discontinuity of the function g(z)=Arg(z^2).

    I have no idea what to do with this problem. Help!
    You need to write this function as \displaystyle u(x, y) + i\,v(x, y). So

    \displaystyle \begin{align*} z^2 &= \left(x + y\,i \right)^2 \\ &= x^2 - y^2 + 2xy\,i \\ \arg{\left(z^2\right)} &= \arctan{ \left( \frac{2xy}{x^2 - y^2} \right) }  + n\pi \textrm{ where }n \in \mathbf{Z} \end{align*}

    Where is this discontinuous?
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    Re: Complex Analysis - Points of Discontinuity

    I can't make the leap from

    =x^2-y^2+2xyi to

    =arctan \frac{2xy}{x^2-y^2}+n\pi where n \in \mathbb{Z}
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    Re: Complex Analysis - Points of Discontinuity

    Quote Originally Posted by tarheelborn View Post
    I can't make the leap from
    =x^2-y^2+2xyi to
    =arctan \frac{2xy}{x^2-y^2}+n\pi where n \in \mathbb{Z}
    If you will note that this question is about the principle value of the argument of a complex number z=a+bi not on any axis is found by the following.
    Arg(z) = \left\{ {\begin{array}{rl} {\arctan \left( {\frac{b}{a}} \right),} & {a > 0}  \\ {\arctan \left( {\frac{b}{a}} \right) + \pi ,} & {a < 0\;\& \,b > 0}  \\ \\ {\arctan \left( {\frac{b}{a}} \right) - \pi ,} & {a < 0\;\& \,b < \pi }  \\ \end{array} } \right.

    For any real number r then Arg(r) = \left\{ {\begin{array}{rl}   {0,} & {r \geqslant 0}  \\   {\pi ,} & {r < 0}  \\ \end{array} } \right.;
    And Arg(ri) = \left\{ {\begin{array}{rl}   {\frac{\pi }{2},} & {r \geqslant 0}  \\   {\frac{{ - \pi }}{2},} & {r < 0}  \\ \end{array} } \right.

    You will need to make adjustments in the above, in that a=x^2-y^2~\&~b=2xy.
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