= Cayley transform
{disambiguate=complex analysis}
{c}
{title2=$C(z)=(z-i)/(z+i)$}
= Cayley transform
{c}
{synonym}
This <Möbius transformation> maps the <upper half-plane> onto the <unit disk>. For $\operatorname{Im}z>0$, the identity $1-|C(z)|^2=4\operatorname{Im}z/|z+i|^2$ puts the image inside the <unit disk>. Its inverse is $z=i(1+w)/(1-w)$, so the map is a <conformal bijection>. On the extended real boundary it maps onto the <unit circle>. It is the scalar complex-analytic version of a Cayley map, rather than the matrix parametrization used in <Cayley transform (Lie theory)>.
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