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

where

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 ,

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).

See also

Conformal Radius, Transfinite Diameter

This entry contributed by Charles Pooh

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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.

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Logarithmic Capacity

Cite this as:

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

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