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Conformal Radius


Let E be a simply connected compact set in the complex plane. By the Riemann mapping theorem, there is a unique analytic function

 f(z)=alphaz+alpha_0+(alpha_1)/z+(alpha_2)/(z^2)+...
(1)

for alpha>0 that maps the exterior of the unit disk conformally onto the exterior of E and takes infty to infty. The number alpha is called the conformal radius of E and alpha_0 is called the conformal center of E.

The function f(z) carries interesting information about the set E. For instance, alpha is equal to the logarithmic capacity of E and

 E subset {w in C:|w-alpha_0|<=2alpha},
(2)

where the equality holds iff E is a segment of length 4alpha. The Green's function associated to Laplace's equation for the exterior of E with respect to infty is given by

 g(w;infty)=ln|f^(-1)(w)|
(3)

for w in C\E.


See also

Logarithmic Capacity, Radius

This entry contributed by Charles Pooh

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References

Pommerenke, C. Univalent functions. Göttingen, Germany: Vandenhoeck & Ruprecht, 1975.

Referenced on Wolfram|Alpha

Conformal Radius

Cite this as:

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

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