Let $\mathrm{z}=\mathrm{a}+\mathrm{ib}={\mathrm{re}}^{\mathrm{i\theta }}\text{ where }\mathrm{a},\mathrm{b},\mathrm{\theta }\in \mathrm{R}$ and $\mathrm{i}=\sqrt{-1}$ 

then  $\mathrm{r}=\sqrt{\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}=\left|\mathrm{z}\right|$

and  $\mathrm{\theta }={\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)=\arg \left(\mathrm{z}\right)$

now, $|\mathrm{z}{|}^2={\mathrm{a}}^2+{\mathrm{b}}^2=(\mathrm{a}+\mathrm{ib}\left)\right(\mathrm{a}-\mathrm{ib})=\mathrm{z}\overline{\mathrm{z}}$

$\Rightarrow \frac{1}{\mathrm{z}}=\frac{\overline{\mathrm{z}}}{|\mathrm{z}{|}^2}$

and  $\left|{\mathrm{z}}_1{\mathrm{z}}_2{\mathrm{z}}_3\dots .{\mathrm{z}}_{\mathrm{n}}\right|=\left|{\mathrm{z}}_1\right|\left|{\mathrm{z}}_2\right|\left|{\mathrm{z}}_3\right|\dots \left|{\mathrm{z}}_{\mathrm{n}}\right|$

if  $\left|\mathrm{f}\left(\mathrm{z}\right)\right|=1$then f(z) is called unimodular. In this case
f(z) can always be expressed as $\mathrm{f}\left(\mathrm{z}\right)={\mathrm{e}}^{\mathrm{i\alpha }},\mathrm{\alpha }\in \mathrm{R}$

Also, ${\mathrm{e}}^{\mathrm{i\alpha }}+{\mathrm{e}}^{\mathrm{i\beta }}={\mathrm{e}}^{\mathrm{i}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)}2\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$  and 

${\mathrm{e}}^{\mathrm{i\alpha }}-{\mathrm{e}}^{\mathrm{i\beta }}={\mathrm{e}}^{\mathrm{i}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)}2\mathrm{isin}\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$ where  $\mathrm{\alpha },\mathrm{\beta }\in \mathrm{R}$