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Math Help - Complex Proofs

  1. #1
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    Complex Proofs

    Prove:

    (Note the * means there is a bar over it)

    1.) arg z* = -arg z(mod 2Pi)

    2.) arg(z/w) = arg z - arg w (mod 2Pi)

    3.) |z| = 0 if and only if z = 0
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  2. #2
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    Quote Originally Posted by Ideasman View Post
    1.) arg z* = -arg z(mod 2Pi)
    2.) arg(z/w) = arg z - arg w (mod 2Pi)
    3.) |z| = 0 if and only if z = 0
    The first is easy.
    z = r\cos \left( \varphi  \right) + ir\sin \left( \varphi  \right)
    \overline z  = r\cos \left( \varphi  \right) - ir\sin \left( \varphi  \right) <br />
\overline z  = r\cos \left( { - \varphi } \right) + ir\sin \left( { - \varphi } \right)<br />

    For the third one way is trival.
    0 = \left| z \right|^2  = {\mathop{\rm Re}\nolimits} ^2 (z) + {\mathop{\rm Im}\nolimits} ^2 (z)\quad  \Rightarrow \quad {\mathop{\rm Re}\nolimits} (z) = 0 \,\& \,{\mathop{\rm Im}\nolimits} (z) = 0
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