Let E be a simply connected compact set in the complex plane. By the Riemann mapping theorem, there is a unique analytic function f(z) = α z + α_0 + α_1/z + α_2/z^2 + ... for α>0 that maps the exterior of the unit disk conformally onto the exterior of E and takes ∞ to ∞. The number α is called the conformal radius of E and α_0 is called the conformal center of E. The function f(z) carries interesting information about the set E.

logarithmic capacity | radius