A conformal map from $\DD$ to $\HH$

The Cayley transform \[ \boxed{F(z)=i\frac{1+z}{1-z}} \] is a Möbius transformation. For \(|z|<1\), \[ \Im F(z) =\Re\frac{1+z}{1-z} =\frac{1-|z|^2}{|1-z|^2}>0, \] so \(F(\mathbb D)\subset\mathbb H\). Its inverse is \[ F^{-1}(w)=\frac{w-i}{w+i}, \] and for \(\Im w>0\) one has \[ \left|\frac{w-i}{w+i}\right|<1. \] Hence \(F\) is a biholomorphism from \(\mathbb D\) onto \(\mathbb H\).