On 10 Nov 2012, at 08:07, MaxJ <email@example.com> wrote:
> Thanks a lot to all you folks for your superb & enlightening comments and solutions. > > QUESTION: > > Based on your solutions, shall I conclude that the lens region can only be mapped into half of unit disk? > > Thanks. > > Max. >
It depends on what sort of mapping you are asking for. The lens can be mapped conformally onto the unit disk, of course. In fact, by the Riemann mapping theorem any open simply connected region (except the entire complex plane) can be mapped conformally onto the unit disk. However, such a mapping cannot be a Moebius transformation (which is the same as a "linear fractional transformation" and which is what you explicitly asked for in your question). A Moebius transformation mapps (generalized) circles to circles and hence it clearly cannot map a lens onto a disk. But of course you can map a lens into a unit disk be means of a Moebius transformation in lots of different ways. In fact, since a Moebius transformation is completely determined by choosing three distinct points in the source space and three distinct points as their images in the target space, the method I posted makes it possible to find all ways of mapping a lens into the unit disk by a Moebius transformation. I chose the Moebius map that sends the lens onto a half disk only because it seemed the most "attractive" of an infinitely many "into" Moebius mappings. Of course none of these mappings is onto. The the biholomorphic mapping which I constructed in another post is a rational map of degree 2 - hence not a Moebius map.