A Möbius transformation is a linear fractional transformation of the Riemann sphere,
given by 
,
where 
are complex numbers with 
. For 
, 
and 
; for 
, 
(Krantz 1999, pp. 82-84).
Let 
and 
,
then
 |
is a Möbius transformation, where 
is the complex conjugate
of 
.
is a conformal mapping self-map
of the open unit disk 
and extends to map the boundary
of the unit disk to itself. The same holds for 
.
Any holomorphic bijection of the unit disk to itself is a composition
of a Möbius transformation with a rotation, and
any such bijection 
can be written as
 |
for some 
and some complex number 
with 
(Krantz 1999, p. 81).
The Smith chart is an engineering application of a Möbius transformation: the map 
sends the open right half of the normalized
impedance plane to the unit disk of reflection coefficients.
