Question

If z is a complex number of unit modulus and argument $\theta$, find arg $\left(\frac{1+z}{1+\overline{z}}\right)$

Answer 1

We have, |z| = 1 and arg (z) = $\theta$

We know that, $|z{|}^2=z\overline{z}$

$\Rightarrow 1=z\overline{z}$

$\Rightarrow .\overline{z}=\frac{1}{z}$

$\therefore arg\left(\frac{1+z}{1+\overline{z}}\right)=arg\left(\frac{1+z}{1+\frac{1}{z}}\right)[\because \overline{z}=\frac{1}{z}]$

=$arg\left[\frac{z(1+z)}{1+z}\right]=arg\left(z\right)=\theta$