Let and , then
is a Möbius transformation, where is the complex conjugate of . is a conformal mapping self-map of the unit disk for each , and specifically of the boundary of the unit disk to itself. The same holds for .
Any conformal self-map of the unit disk to itself is a composition of a Möbius transformation with a rotation, and any conformal self-map of the unit disk can be written in the form
for some Möbius transformation and some complex number with (Krantz 1999, p. 81).