Conformal Radius

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 .

REFERENCES:

Pommerenke, C. Univalent functions. Göttingen, Germany: Vandenhoeck & Ruprecht, 1975.