= Möbius transformations from the real axis to the unit circle
{c}
{title2=$w=\lambda(z+k)/(z+\overline k)$}
A nonconstant <Möbius transformation> mapping the <real line> into the <unit circle> has the displayed form with $|\lambda|=1$ and $k$ nonreal. Indeed $|ax+b|^2=|cx+d|^2$ for real $x$ yields $|a|=|c|\ne0$, $|b/a|=|d/c|$ and $\operatorname{Re}(b/a)=\operatorname{Re}(d/c)$. Thus $d/c$ is $b/a$ or its <complex conjugate>; the first possibility makes the map constant, so the second is necessary. Conjugate numerator and denominator have equal modulus on the <real line>, proving the converse as well.
Back to article page
