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 
.
