Let \(\displaystyle \varepsilon\) the Stereographic projection of \(\displaystyle \mathbb{S}^1- \{i \}\), that is

\(\displaystyle \varepsilon : \mathbb{S}^1 - \{i \} \longrightarrow{\mathbb{R}}\)

where

\(\displaystyle \varepsilon(z)=\displaystyle\frac{RE(z)}{1-Im(z)}\)

Find a formula for \(\displaystyle \varepsilon^{-1} :\mathbb{R}\rightarrow{\mathbb{S}^1}- \{i \} \)

I tried to take the line that passes through i and a point on the real axis to see where the circle intersects but I can not find the formula

Thanks