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 .