# Linear fractional transformation

1. Mar 19, 2012

### arthurhenry

The following comes from the complex analysis text by Joseph Bak:

He is trying to determine all conformal mappings f of upper half plane H onto the unit disk.

"Let us first assume that f is an LFT and f(a)=0 for Im(a)>0.
Then, since the real axis is mapped into the unit circle, it follows by Schwarz Principle of reflection that f(a*)=Infinity, so that...

Here a* denotes the conjugate of a.

I am having difficulty following how the Schwarz Reflection Principle is applied...and we get that f(a*)=Infinity.

thank you.

Also, Isn't true that the inverse Caylet Transform map the unit circle into real axis.If so, why "there is no non-constant analytic function in the unit disk that is real valued on the unit circle" ?
Thank you