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conformal radius
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(Definition)
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Example 1 For example, take $G = B(0,\delta)$ (the open ball of radius $\delta$ around 0) for some $\delta > 0$ , then $r(G,0) = \delta$ because we have a map $f(z) = \delta \cdot z$ as our unique map. And thus this definition coincides with our normal definition of radius for this special case.
Example 2 For another example we look at how the conformal radius is affected by the choice of the point $a$ . So suppose that we take $G$ to be the unit disc (${\mathbb{D}}$ ) itself and we take some point $a \in {\mathbb{D}}$ . The unique map that takes 0 to $a$ is the map $f(z) = \frac{z+a}{1+\bar{a}z}$ (where $\bar{a}$ is the complex conjugate of $a$ ) and by the quotient rule we get that $f'(z) = \frac{1-\lvert a \rvert^2}{(1+\bar{a}z)^2}$ . And so $r({\mathbb{D}},a) = f'(0) =
1-\lvert a \rvert^2$ , so the conformal radius of the unit disc goes to 0 as we move the point $a$ towards the boundary of the disc, and it is largest (equal to 1) when $a = 0$ .
From the first example we can now see another way of characterizing the conformal radius. Take the inverse map (inverses of holomorphic one-to-one functions are also always holomorphic) and call it $\varphi \colon G \to {\mathbb{D}}$ (the map such that $\varphi (f(z)) = z$ ). We take the derivative (see the entry on univalent functions) we get $\varphi'(f(0)) = \frac{1}{f'(0)}$ , that is $\varphi'(a) = \frac{1}{r}$ (where we call $r = r(G,a)$ for
brevity now). If we multiply the map by the conformal radius we get a map $\gamma\colon G \to B(0,r)$ such that $\gamma (z) = r \cdot \varphi(z)$ and $\gamma'(a) = 1$ . By uniqueness of the map arising from the Riemann mapping theorem we can see that $\gamma$ is also unique. Thus we could define the conformal radius as follows.
Definition 2 Let $G \subset {\mathbb{C}}$ be a region and let $a \in G$ be any point. By application of Riemann mapping theorem there exists a unique map $\gamma \colon G \to B(0,r)$ for some $r > 0$ , such that $\gamma(a) = 0$ and $\gamma'(a) = 1$ . The conformal radius is then defined as $r(G,a) = r$ .
This definition gives more of an intuitive understanding of why we'd call this the conformal radius of $G$ . We look at the unique map with $\gamma'(a) = 1$ , that is, the map that doesn't ``stretch'' the set. So the radius of $G$ with respect to $a$ is really the radius of the unique ball around zero to which $G$ is conformally equivalent without any ``stretching'' needed.
- 1
- S. Rohde, M. Zinsmeister. Variation of the conformal radius, Journal d'Analyse (to appear). Available at http://www.math.washington.edu/~rohde/papers/rozi.ps
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"conformal radius" is owned by jirka.
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Cross-references: conformally equivalent, ball, application, derivative, functions, inverse, disc, boundary, quotient rule, complex conjugate, radius, open ball, unit disc, map, holomorphic, onto, one-to-one, Riemann mapping theorem, point, plane, region, simply connected
This is version 6 of conformal radius, born on 2004-04-15, modified 2005-03-05.
Object id is 5771, canonical name is ConformalRadius.
Accessed 2213 times total.
Classification:
| AMS MSC: | 30C55 (Functions of a complex variable :: Geometric function theory :: General theory of univalent and multivalent functions) |
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Pending Errata and Addenda
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