Transform Name for $(1-z)/(1+z)$

Question

I am trying to find the actual name of the transform $f(z) = \frac{1-z}{1+z}$; the transform from the open unit disk to the right half plane. Another variant; the transform from the upper half plane to the unit disk, eg $f(z) = \frac{z-i}{z+i}$ is also often used/cited.

I know it's a Möbius map and I'm quite comfortable with the map and its properties but I could have sworn it had an official name; the "(insert mathematician name here)'s transform" or something along those lines. My Google-fu is failing me, and none of my books at hand have any names associated to it.

Am I crazy and misremembering, and this is just one (of many) Möbius maps? Or does this specific transform have a name? So far all I've found is the Z-transform (which is clearly not it) and Möbius map, which is also not what I'm thinking of.

Answer 1

The map $$\frac{z-i}{z+i}$$ is called the Cayley transform (and it generalizes to operators). Its inverse is $i$ times $1$ over your map, so I suppose that you could call your map $i$ times $1$ over the inverse of the Cayley transform.