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Math Help - Complex Analysis- help needed with proof

  1. #1
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    Complex Analysis- help needed with proof

    Complex Argument: how do I show that

    arg \overline{z} = - arg z. If z is not a real number
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  2. #2
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    Re: Complex Analysis- help needed with proof

    if we write z = r(\cos\theta + i\sin\theta)

    then \text{arg}(z) = \theta

    now \overline{z} = \overline{r(\cos\theta + i\sin\theta)} = \overline{r}\overline{(\cos\theta + i\sin\theta)}

    =r(\cos\theta - i\sin\theta) = r(\cos(-\theta) + i\sin(-\theta))

    so \text{arg}(\overline{z}) = -\theta = -\text{arg}(z)
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Complex Analysis- help needed with proof

    Another way (specially if the OP means the principal argument). Denoting \arg z the principal argument of z\neq 0 we have \arg(z_1z_2)=\arg z_1+\arg z_2 if z_1z_2\neq 0 and -\pi<\arg z_1+\arg z_2\leq \pi . These conditions are verified for z_1=z and z_2=\bar{z} and 0\neq z\not\in\mathbb{R} . So z\bar{z}=|z|^2 which implies \arg z+\arg\bar{z}=0 i.e. \arg\bar{z}=-\arg{z} .
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