Hello all,


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.


Greg



General form of Schwarz reflection principle
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 .


