it doesn't seem to match your function. Do you want points near center to map to infinity or to the edge of a circle with radius 1/R?
Here's a pic of map I'm thinking about... blue x's map to red o's... seems like people would have studied its props https://twitter.com/vgr/status/794994887254941696 …pic.twitter.com/ycetoHd6Gm
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but image looks like simple example of conformal mapping, angles are preserved
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