# Complex Analysis- help needed with proof

• Dec 24th 2011, 07:24 AM
flaming
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
• Dec 24th 2011, 07:51 AM
Deveno
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 $\text{arg}(\overline{z}) = -\theta = -\text{arg}(z)$
• Dec 24th 2011, 08:47 AM
FernandoRevilla
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. $\arg\bar{z}=-\arg{z}$ .