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).
