If you are interested in conformal mapping you can read the entire detailed thread

here.

$\displaystyle \frac{i-z}{i+z} = -\frac{z-i}{z+i} = - \frac{z+i-2i}{z+i} = -1 + \frac{2i}{z+i} $

Let $\displaystyle f: z\mapsto z+i$, let $\displaystyle g:z\mapsto \tfrac{1}{z}$, let $\displaystyle h: z \mapsto i$, let $\displaystyle F: z\mapsto 2z$, let $\displaystyle G:z\mapsto z-1$. Thus, $\displaystyle f$ is translation, $\displaystyle g$ is inversion, $\displaystyle h$ is rotation (by $\displaystyle \tfrac{\pi}{2}$), $\displaystyle F$ is dilation, and $\displaystyle G$ is translation.

However, $\displaystyle G\circ H \circ h\circ g \circ f: z\mapsto \tfrac{i-z}{i+z}$.