General form of Schwarz reflection principle

Question

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

Answer 1

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 .

Answer 2

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 $z = \infty$.