Logarithmic Capacity
المؤلف:
Hille, E
المصدر:
Analytic Function Theory. New York: Chelsea, 1973.
الجزء والصفحة:
...
1-11-2018
753
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 
,
![lim_(n->infty)max_(<span style=]() {w_1,...,w_n} subset E)(product_(1<=j<k<=n)|w_j-w_k|)^(2/[n(n-1)]). " src="http://mathworld.wolfram.com/images/equations/LogarithmicCapacity/NumberedEquation3.gif" style="height:51px; width:234px" /> |
(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.
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