TOPICS
Search

Logarithmic Capacity


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

 gamma(E)=e^(-V(E)),
(1)

where

 V(E)=inf_(nu)int_(E×E)ln1/(|u-v|)dnu(u)dnu(v),
(2)

and nu runs over each probability measure on E. The quantity V(E) is called the Robin's constant of E and the set E is said to be polar if V(E)=+infty or equivalently, gamma(E)=0.

The logarithmic capacity coincides with the transfinite diameter of E,

 lim_(n->infty)max_({w_1,...,w_n} subset E)(product_(1<=j<k<=n)|w_j-w_k|)^(2/[n(n-1)]).
(3)

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


See also

Conformal Radius, Transfinite Diameter

This entry contributed by Charles Pooh

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Logarithmic Capacity

Cite this as:

Pooh, Charles. "Logarithmic Capacity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LogarithmicCapacity.html

Subject classifications