Cayley map

The Cayley map is a conformal map that sends the upper half-plane to the unit disk.
It is given by

Key properties

• Domain and codomain:
  $\mathbb{H}=\{z\in \mathbb{C}:\mathrm{\Im }z>0\}⟶\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$.
• Boundary correspondence:
  The real line $\mathbb{R}$ maps onto the unit circle $\partial \mathbb{D}$.
• Inverse map:

• Special values:
  • $\phi (i)=0$
  • $\phi (0)=-1$
  • $\phi (\mathrm{\infty })=1$
  • $\phi (1)=-i$
  • $\phi (-1)=i$