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).
The Smith chart is an engineering application of a Möbius transformation: the map 
sends the right half of the normalized impedance
plane to the unit disk of reflection coefficients.
