The conformal radius of a simply connected compact set 
in the complex plane is the
number 
in the unique analytic function
 |
(1)
|
with 
that, by the Riemann mapping theorem, maps
the exterior of the unit disk conformally
onto the exterior of 
and takes 
to 
.
The number 
is called the conformal center of 
. This radius of 
is determined by the normalized mapping above.
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 
.
