General form of Schwarz reflection principle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:58:04Z http://mathoverflow.net/feeds/question/80430 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80430/general-form-of-schwarz-reflection-principle General form of Schwarz reflection principle Greg Markowsky 2011-11-08T21:54:36Z 2011-11-29T16:32:13Z <p>Hello all,</p> <p>It is easy to find results on reflecting holomorphic functions over circles and lines, but I am wondering what there is for reflecting over smooth curves, or analytic arcs, etc. In particular, I am interested in the conformal map f from the upper half-plane to $\{x+yi : y>1/(1+x^2)\}$ which maps $0$ to $i$ and fixes infinity (so, say, maps $i$ to $2i$). It seems to me that I should be able to extend $f$ to be analytic in a neighborhood of infinity, but I cannot find a reference. Any help will be appreciated.</p> <p>Greg</p> http://mathoverflow.net/questions/80430/general-form-of-schwarz-reflection-principle/80837#80837 Answer by Mohan Ramachandran for General form of Schwarz reflection principle Mohan Ramachandran 2011-11-13T18:06:29Z 2011-11-13T18:06:29Z <p>For reflection across analytic arcs see Caratheodory's book Conformal Representation pages 87-90 or his book Theory of Functions vol 2 pages 101-104 .</p> http://mathoverflow.net/questions/80430/general-form-of-schwarz-reflection-principle/82192#82192 Answer by Robert Bryant for General form of Schwarz reflection principle Robert Bryant 2011-11-29T16:32:13Z 2011-11-29T16:32:13Z <p>For your specific question, note that the domain you describe $\mathbb{D}$, consisting of those $z = x+iy$ for which $y > 1/(1+x^2)$, when regarded as a domain in the extended complex plane, $\mathbb{CP}^1= \mathbb{C}\cup\infty$, is a disk with a smooth, real-analytic boundary in $\mathbb{CP}^1$, so the mapping you are describing does, in fact, extend analytically across the boundary, everywhere along the boundary. In particular, it extends analytically (meromorphically, actually) to a neighborhood of $z = \infty$.</p>