Let 
be a simply connected compact
set in the complex plane. By the Riemann
mapping theorem, there is a unique analytic
function
 |
(1)
|
for 
that maps the exterior of the unit disk conformally
onto the exterior of 
and takes 
to 
.
The number 
is called the conformal radius of 
and 
is called the conformal center of 
.
The function 
carries interesting information about the set 
. For instance, 
is equal to the logarithmic
capacity of 
and
 |
(2)
|
where the equality holds iff 
is a segment of length 
. The Green's function
associated to Laplace's equation for the exterior
of 
with respect to 
is given by
 |
(3)
|
for 
.
