Impedance matching without Smith Chart

The Smith Chart is a mapping between reflection coefficient and normalized impedance characterized by a pair of equations:

Γ = \frac{z - 1}{z + 1}

z = \frac{Γ + 1}{1 - Γ}

The impedance matching problem is mainly finding the $z$ that having reflection coefficient $Γ_{z} = ℜ_{z} + j ℑ_{z}$ satisfies:

\begin{aligned} ℜ_{z}^{2} + ℑ_{z}^{2} & = Γ^{2} \\ (ℜ_{z} - \frac{1}{2})^{2} + ℑ_{z}^{2} & = \frac{1}{4} \end{aligned}

Using some algebra it is easy to find out $ℜ_{z} = Γ^{2}$ , $ℑ_{z} = \sqrt{Γ^{2} - Γ^{4}}$ . Now, we already know that the real component of $z$ is $1$ . Then the imaginary part of the two matching impedance would be:

\pm \frac{2 ℑ_{z}}{ℜ_{z}^{2} - 2 ℜ_{z} + ℑ_{z}^{2} + 1}