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 
,
![lim_(n->infty)max_({w_1,...,w_n} subset E)(product_(1<=j<k<=n)|w_j-w_k|)^(2/[n(n-1)]).](/images/equations/LogarithmicCapacity/NumberedEquation3.svg) |
(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).
