To compute these neighborhoods, find a conformal mapping that takes a real interval (or the entire real axis) to the circle, and then map the image of a grid containing the domain. (This of course can be generalized with maps to any curve) 2/3
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Conformal mappings for the given pictures are normalized Jacobi sn, inverse stereoscopic, cos+i sin, and a frequency modulated version of cos+i sin. These were rendered in online Mathematica and have some artifacts. 3/3
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