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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 $\{x+yi : y>1/(1+x^2)y>1/(1+x^2)\}$ which maps 0 $0$ to i $i$ and fixes infinity(so, infinity (so, say, maps i $i$ to 2i). $2i$). It seems to me that I should be able to extend f $f$ to be analytic in a neighborhood of infinity, but I cannot find a reference. Any help will be appreciated.

Greg