Math Help - Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

1. Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

Let Arg(w) denote that value of the argument between -π and π (inclusive). Show
that:

Arg[(z-1)/(z+1)] = { π/2, if Im(z) > 0 or -π/2 ,if Im(z) < 0.

where z is a point on the unit circle ∣z= 1.

My attempt: I know
Arg[(z-1)/(z+1)] = Arg(z-1) - Arg(z+1) and the arg(w) = arctan(b/a), but im stuck.

2. Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

Originally Posted by habsfan31
Let Arg(w) denote that value of the argument between -π and π (inclusive). Show
that: Arg[(z-1)/(z+1)] = { π/2, if Im(z) > 0 or -π/2 ,if Im(z) < 0.
where z is a point on the unit circle ∣z[/FONT]
Know that $\frac{z-1}{z+1}=\frac{(z-1)(\overline{z}+1)}{|z+1|^2}=\frac{z\overline{z}+z-\overline{z}-1}{|z+1|^2}=\frac{2\text{Im}(z)i}{|z+1|^2}$

3. Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

Originally Posted by Plato
Know that $\frac{z-1}{z+1}=\frac{(z-1)(\overline{z}+1)}{|z+1|^2}=\frac{z\overline{z}+z-\overline{z}-1}{|z+1|^2}=\frac{2\text{Im}(z)i}{|z+1|^2}$
I understand what you did, but not sure why you did it and what im supposed to do next.

4. Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

Originally Posted by habsfan31
but not sure why you did it and what im supposed to do next.
I did it to answer the question: $\text{Arg}\left(\frac{z-1}{z+1}\right)=~?$
Do you understand that $\text{Arg}(z)$ is a real number?
Do you understand that $-\pi<\text{Arg}(z)\le\pi~?$
Do you understand that if $r\in\mathbb{R}$ then $\text{Arg}(ri)=\pm\frac{\pi}{2}~?$

Because $\frac{z-1}{z+1}=\frac{2\text{Im}(z)i}{|z+1|^2}$ and $\text{Im}\left(\frac{2\text{Im}(z)i}{|z+1|^2} \right)=\frac{2\text{Im}(z)}{|z+1|^2}\in\mathbb{R}$
then $\text{Arg}\left(\frac{z-1}{z+1}\right)=\pm\frac{\pi}{2}$

5. Re: Complex Analysis - Value of Arg[(z-1)/(z+1)] between -pi and pi

Got it thanks!