For example, take (the open ball of radius around 0) for some , then because we have a map as our unique map. And thus this definition coincides with our definition of radius for this special case.
For another example we look at how the conformal radius is affected by the choice of the point . So suppose that we take to be the unit disc () itself and we take some point . The unique map that takes 0 to is the map (where is the complex conjugate of ) and by the quotient rule we get that . And so , so the conformal radius of the unit disc goes to 0 as we move the point towards the boundary of the disc, and it is largest (equal to 1) when .
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 (the map such that ). We take the derivative (see the entry on univalent functions (http://planetmath.org/UnivalentFunction)) we get , that is (where we call for brevity now). If we multiply the map by the conformal radius we get a map such that and . By uniqueness of the map arising from the Riemann mapping theorem we can see that is also unique. Thus we could define the conformal radius as follows.
Let be a region and let be any point. By application of Riemann mapping theorem there exists a unique map for some , such that and . The conformal radius is then defined as .
This definition gives more of an intuitive understanding of why we’d call this the conformal radius of . We look at the unique map with , that is, the map that doesn’t “stretch” the set. So the radius of with respect to is really the radius of the unique ball around zero to which is conformally equivalent without any “stretching” needed.
- 1 S. Rohde, M. Zinsmeister. , Journal d’Analyse (to appear). Available at http://www.math.washington.edu/ rohde/papers/rozi.pshttp://www.math.washington.edu/ rohde/papers/rozi.ps
|Date of creation||2013-03-22 14:18:33|
|Last modified on||2013-03-22 14:18:33|
|Last modified by||jirka (4157)|