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


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},

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


for w in C\E.

See also

Logarithmic Capacity, Radius

This entry contributed by Charles Pooh

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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.

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