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.


1 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.

  • $\begingroup$ Ah HA! This is it, the Cayley transform was the one I was thinking about. Thanks, this has been bugging me for way longer than I want to admit. Answer accepted (or will be when the timer lets me). $\endgroup$
    – Jason
    Commented Jun 16, 2019 at 21:07
  • $\begingroup$ @Jason No problem, I'm glad it helped! $\endgroup$
    – cmk
    Commented Jun 16, 2019 at 21:07

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