Logarithmic Capacity

The logarithmic capacity of a compact set  in the complex plane is given by

(1)

where

(2)

and  runs over each probability measure on . The quantity  is called the Robin's constant of  and the set  is said to be polar if  or equivalently, .

The logarithmic capacity coincides with the transfinite diameter of ,

(3)

If  is simply connected, the logarithmic capacity of  is equal to the conformal radius of . Tables of logarithmic capacities have been calculated (e.g., Rumely 1989).

REFERENCES:

Hille, E. Analytic Function Theory. New York: Chelsea, 1973.

Rumely, R. Capacity Theory on Algebraic Curves. New York: Springer-Verlag, pp. 348-351, 1989.